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➕ Algebra 1 · Exponents and Polynomials

Adding and Subtracting Polynomials

Add and subtract polynomials by combining like terms in horizontal and vertical form, dodge the sign trap, and check with a test value, with diagrams and practice.

Grades 9, 10⏱ 15 min✔ AI-checkedCCSS A-APR.A.1CCSS A-SSE.A.1a
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
  • A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

By the end you can…

  • Name polynomials by their number of terms (monomial, binomial, trinomial) and give the degree and leading coefficient.
  • Write polynomials in standard form and add them in horizontal and vertical form by combining like terms.
  • Subtract polynomials by distributing the negative to every term of the second polynomial.
  • Explain why the sum or difference of two polynomials is again a polynomial, and check answers with a test value.
  • Use polynomial addition and subtraction to find perimeters and areas.

Lesson (school style)

A polynomial is a sum of terms, where each term is a number times a variable (or several variables) raised to whole-number exponents. Adding polynomials is just combining like terms. Subtracting is the same job with one extra step: the minus sign in front of the second polynomial flips the sign of every term inside it. This lesson builds on Combining Like Terms and the Distributive Property, and you will check your answers with a test value.

Words for polynomials

A term is one piece of an expression, separated from the others by plus or minus signs. A monomial is a single term that is a number, a variable, or a number times variables with whole-number exponents, like 7, −3x or 5x2. A polynomial is a monomial or a sum of monomials. A difference counts too, because subtracting is the same as adding a negative term.

Polynomials with few terms have special names:

  • Monomial: 1 term, like 6x3.
  • Binomial: 2 terms, like 4x − 9.
  • Trinomial: 3 terms, like x2 + 5x + 6.

A polynomial with 4 or more terms is just called a polynomial. These names count the terms after like terms have been combined.

The coefficient of a term is the number multiplied by its variable part: −3 in −3x, and 1 in x2. The degree of a term with one variable is its exponent. So 5x2 has degree 2, −3x has degree 1, and a nonzero constant like 7 has degree 0. If a term has more than one variable, add the exponents: 3x2y has degree 2 + 1 = 3. The degree of a polynomial is the highest degree among its terms. For example, 3x + 8x5 − x2 has degree 5, even though 3x is written first.

Not every expression is a polynomial. The exponents on the variables must be whole numbers (0, 1, 2, 3, ...). So 5/x (which is 5x−1) and √x (which is x1/2) are not polynomials. The number 0 by itself is a polynomial, called the zero polynomial, and its degree is left undefined.

Standard form

For a polynomial in one variable, standard form writes the terms in descending order of degree: the highest power first and the constant last. For example, x − 6x3 + 2 + 9x2 in standard form is −6x3 + 9x2 + x + 2. Move each term together with the sign in front of it.

The first term is the leading term, and its coefficient is the leading coefficient (−6 in the example). For a polynomial in one variable written in standard form, the degree is the exponent of the leading term. Standard form makes like terms easy to find, and it makes answers easy to compare.

This is a different use of the words “standard form” from the Standard Form of a Line, Ax + By = C.

Adding polynomials

Adding polynomials means combining like terms. Like terms have exactly the same variable part: the same letters with the same exponents. Add the coefficients and keep the variable part, as in Combining Like Terms.

Horizontal form. Adding does not change any signs, so you can drop the parentheses, group like terms, and combine:

(2x2 + 5x) + (3x2 − 2x + 4) = 2x2 + 3x2 + 5x − 2x + 4 = 5x2 + 3x + 4

Vertical form. Write one polynomial under the other in standard form, with like terms in the same column, then add each column. If a power is missing, write a placeholder 0 so the columns stay lined up. For the sum above, the columns are:

  • x2 column: 2 + 3 = 5
  • x column: 5 + (−2) = 3
  • constant column: 0 + 4 = 4 (the first polynomial has no constant term, so it gets a 0)

Both methods give the same answer. Horizontal form is quick for short polynomials, and vertical form helps keep long ones organized.

Subtracting polynomials: flip every sign

A minus sign in front of a polynomial applies to all of it. By the distributive property, −(4x2 − 7x + 3) = (−1)(4x2 − 7x + 3), so every term inside gets multiplied by −1 and every sign flips:

−(4x2 − 7x + 3) = −4x2 + 7x − 3

So subtracting is the same as adding the opposite: A − B = A + (−B), where −B is B with every sign changed.

The classic trap is flipping only the first term. Take (9x − 2) − (4x − 7):

  • Correct: 9x − 2 − 4x + 7 = 5x + 5.
  • Trap: 9x − 2 − 4x − 7 = 5x − 9. The −7 was left alone, but it should have become +7.

A check at x = 3 catches the trap. The original is 25 − 5 = 20, and the correct answer gives 5(3) + 5 = 20. The trap answer gives 5(3) − 9 = 6.

In vertical form, write the polynomial being subtracted underneath, change every sign in that bottom row, and then add the columns.

The answer is a polynomial again

When you add or subtract polynomials, each new coefficient is a + b or a − b, and the exponents stay whole numbers. So the result is always a polynomial. We say polynomials are closed under addition and subtraction. Integers behave the same way: the sum or difference of two integers is an integer, like 3 − 8 = −5. That is the idea behind the standard A-APR.A.1, which says polynomials form a system analogous to the integers. The next lesson shows that multiplying also stays inside the system. Dividing does not: 1 ÷ x (for x ≠ 0) is not a polynomial.

The degree of the answer is never bigger than the larger of the two degrees, but it can be smaller, because leading terms can cancel. For example, (x3 + 4) − (x3 − 6) = 10. The x3 terms cancel, so the answer is a constant, which is still a polynomial (with degree 0). If everything cancels, as in (x2 + 1) − (x2 + 1) = 0, the answer is the zero polynomial.

Perimeter and area

Perimeter is the distance around a shape, so you add the side lengths. If the sides are polynomials, add the polynomials. If you know the perimeter and all but one side, subtract the known sides from the perimeter.

Area works the same way for shapes made of parts. The total area is the sum of the areas of the parts, and the area left after cutting out a piece is a difference. In this lesson the part areas are given to you as polynomials. In Multiplying Polynomials you will learn to find the area of a rectangle with polynomial sides.

Keep the units. If x is a length in meters, perimeters are in meters and areas are in square meters.

Check with a test value

To check your work, put the same number in for x in the original expression and in your answer. The two results should match. If they differ, there is a mistake.

Choose a value such as x = 3, and avoid 0, 1 and 2, because they can hide mistakes:

  • At x = 0, every term with an x becomes 0, so only the constants get tested.
  • At x = 1, every power of x equals 1, so mistakes that mix up powers can hide. Suppose the right answer is 3x2 + 4x and you wrote 4x2 + 3x. Both equal 7 at x = 1. At x = 3 they are 39 and 45, so the mistake shows.
  • At x = 2, a mistake like writing 2x2 where 4x belongs can hide, because both equal 8.

Every single value has some blind spot. At x = 3, for example, x2 and 3x are both 9. So matching at one value is a good sign, not a proof. For extra confidence, test a second value, such as x = 4 or x = −2.

Key words

Term
One piece of an expression, separated from the others by plus or minus signs. It keeps the sign in front of it: in 5x2 − 3x + 1 the terms are 5x2, −3x and 1.
Monomial
A polynomial with one term: a number, a variable, or a number times variables with whole-number exponents, like −3x or 5x2.
Binomial
A polynomial with exactly two terms, like 4x − 9.
Trinomial
A polynomial with exactly three terms, like x2 + 5x + 6.
Polynomial
A monomial or a sum of monomials. The exponents on the variables are whole numbers (0, 1, 2, ...).
Coefficient
The number multiplied by the variable part of a term: −3 in −3x, and 1 in x2.
Degree
For a term with one variable, its exponent. For a polynomial, the highest degree among its terms. (For a term with several variables, add the exponents.)
Standard form
A polynomial in one variable written with its terms in descending order of degree, highest power first and the constant last, like −6x3 + 9x2 + x + 2.
Like terms
Terms with exactly the same variable part: the same letters with the same exponents. Only like terms can be combined.
Opposite of a polynomial
The polynomial with the sign of every term changed. The opposite of 4x2 − 7x + 3 is −4x2 + 7x − 3. Subtracting a polynomial is the same as adding its opposite.

Worked examples

Example 1. Write 5 − 2x3 + 4x in standard form. Then say what type of polynomial it is by its number of terms, and give its degree and leading coefficient.

  1. List the terms with their signs: +5, −2x3 and +4x.
  2. Order them from the highest power to the lowest, moving each sign with its term: −2x3, then 4x, then 5.
  3. Standard form: −2x3 + 4x + 5.
  4. It has 3 terms, so it is a trinomial. With one variable, the degree is the highest exponent, which is 3.
  5. The leading coefficient is the coefficient of the leading term −2x3, which is −2. It is not 5, the term that was written first.

Answer: Answer: −2x3 + 4x + 5: a trinomial of degree 3 with leading coefficient −2

Example 2. Add (4x2 + 3x − 7) + (2x2 − 5x + 9).

  1. Adding does not change any sign, so remove the parentheses: 4x2 + 3x − 7 + 2x2 − 5x + 9.
  2. Group like terms, keeping each sign with its term: (4x2 + 2x2) + (3x − 5x) + (−7 + 9).
  3. Combine the coefficients: 4 + 2 = 6, then 3 + (−5) = −2, then −7 + 9 = 2.
  4. Write the result in standard form: 6x2 − 2x + 2.
  5. Check with x = 3: the first polynomial is 36 + 9 − 7 = 38 and the second is 18 − 15 + 9 = 12, so the sum is 38 + 12 = 50. The answer gives 6(9) − 2(3) + 2 = 54 − 6 + 2 = 50.

Answer: Answer: 6x2 − 2x + 2

Example 3. Add in vertical form: (3x3 + 5x − 2) + (x3 − 4x2 + 6x + 8).

  1. Write both polynomials in standard form and stack them with like terms in the same column. The first polynomial has no x2 term, so give it a placeholder: 3x3 + 0x2 + 5x − 2.
  2. Add each column. x3: 3 + 1 = 4. x2: 0 + (−4) = −4. x: 5 + 6 = 11. Constants: −2 + 8 = 6.
  3. Read across: 4x3 − 4x2 + 11x + 6.
  4. Check with x = 3: the first polynomial is 81 + 15 − 2 = 94 and the second is 27 − 36 + 18 + 8 = 17, so the sum is 111. The answer gives 108 − 36 + 33 + 6 = 111.

Answer: Answer: 4x3 − 4x2 + 11x + 6

Example 4. Subtract (7x2 + 3x − 4) − (2x2 − 5x + 6).

  1. The minus sign applies to every term of the second polynomial, so flip all three signs: −(2x2 − 5x + 6) = −2x2 + 5x − 6.
  2. Rewrite without parentheses: 7x2 + 3x − 4 − 2x2 + 5x − 6.
  3. Group like terms: (7x2 − 2x2) + (3x + 5x) + (−4 − 6).
  4. Combine: 5x2 + 8x − 10.
  5. Check with x = 3: the first polynomial is 63 + 9 − 4 = 68 and the second is 18 − 15 + 6 = 9, so the difference is 68 − 9 = 59. The answer gives 45 + 24 − 10 = 59.
  6. If only the first sign had flipped, we would get 5x2 − 2x + 2, which is 41 at x = 3. That does not match, so the trap answer is wrong.

Answer: Answer: 5x2 + 8x − 10

Example 5. Subtract (2x3 + 5x) − (2x3 − x + 4).

  1. The second polynomial has the terms 2x3, −x and +4. Flip all three signs: −2x3 + x − 4.
  2. Rewrite: 2x3 + 5x − 2x3 + x − 4.
  3. Group like terms: (2x3 − 2x3) + (5x + x) + (−4). The x3 terms cancel to 0.
  4. The result is 6x − 4. The degree dropped from 3 to 1, and the answer is still a polynomial.
  5. Check with x = 3: the first polynomial is 54 + 15 = 69 and the second is 54 − 3 + 4 = 55, so the difference is 14. The answer gives 6(3) − 4 = 14.

Answer: Answer: 6x − 4

Example 6. Subtract [(3/4)x2 − x + 2] − [(1/4)x2 − 3x − 5].

  1. Flip all three signs of the second polynomial: −(1/4)x2 + 3x + 5.
  2. Rewrite: (3/4)x2 − x + 2 − (1/4)x2 + 3x + 5.
  3. The x2 terms: 3/4 − 1/4 = 2/4 = 1/2. The x terms: −1 + 3 = 2. The constants: 2 + 5 = 7.
  4. Write the result: (1/2)x2 + 2x + 7.
  5. Check with x = 4, which clears the fractions: the first polynomial is (3/4)(16) − 4 + 2 = 10 and the second is (1/4)(16) − 12 − 5 = −13, so the difference is 10 − (−13) = 23. The answer gives (1/2)(16) + 8 + 7 = 23.

Answer: Answer: (1/2)x2 + 2x + 7

Example 7. Add (3x2y + 2xy − 5) + (x2y − 4xy + 1). What is the degree of the answer?

  1. Like terms need the same letters with the same exponents. The x2y terms are like each other, the xy terms are like each other, and the constants are like each other. The terms x2y and xy are not like terms.
  2. The x2y terms: 3 + 1 = 4. The xy terms: 2 + (−4) = −2. The constants: −5 + 1 = −4.
  3. Write the result: 4x2y − 2xy − 4.
  4. The degree of 4x2y is 2 + 1 = 3, and the other terms have lower degree, so the polynomial has degree 3.
  5. Check with x = 3 and y = 2: the first polynomial is 54 + 12 − 5 = 61 and the second is 18 − 24 + 1 = −5, so the sum is 56. The answer gives 4(18) − 2(6) − 4 = 56.

Answer: Answer: 4x2y − 2xy − 4, which has degree 3

Example 8. A student wrote (5x + 4) − (2x − 3) = 3x + 1. Use a test value to decide whether this is right, and find the correct answer.

  1. Pick x = 3 (not 0, 1 or 2).
  2. The original expression gives (5(3) + 4) − (2(3) − 3) = 19 − 3 = 16.
  3. The student’s answer gives 3(3) + 1 = 10. The values differ, so the answer is wrong.
  4. Find the slip: the minus sign was not applied to the −3. Flip both signs: 5x + 4 − 2x + 3 = 3x + 7.
  5. Check the corrected answer at x = 3: 3(3) + 7 = 16, which matches the original.

Answer: Answer: The student’s answer is wrong. The correct answer is 3x + 7.

Example 9. The perimeter of a triangle is (9x + 4) centimeters. Two of its sides are (3x + 1) and (4x − 2) centimeters. Find the third side.

  1. The perimeter is the sum of the three sides, so the third side is the perimeter minus the other two sides.
  2. Write it as (9x + 4) − (3x + 1) − (4x − 2). Flip every sign in each polynomial that is being subtracted: 9x + 4 − 3x − 1 − 4x + 2.
  3. The x terms: 9 − 3 − 4 = 2. The constants: 4 − 1 + 2 = 5.
  4. The third side is (2x + 5) centimeters.
  5. Check by adding all three sides: (3x + 1) + (4x − 2) + (2x + 5) = 9x + 4, which is the perimeter.

Answer: Answer: (2x + 5) centimeters

Example 10. A garden bed has area (3x2 + 5x + 2) square meters. A path across it covers (x2 + 5x + 1) square meters. What area is left for planting?

  1. The planting area is the whole bed minus the path: (3x2 + 5x + 2) − (x2 + 5x + 1).
  2. Flip all three signs of the path’s polynomial and rewrite: 3x2 + 5x + 2 − x2 − 5x − 1.
  3. The x2 terms: 3 − 1 = 2. The x terms: 5 − 5 = 0, so they cancel. The constants: 2 − 1 = 1.
  4. The planting area is (2x2 + 1) square meters.
  5. Check with x = 3: the bed is 27 + 15 + 2 = 44 and the path is 9 + 15 + 1 = 25, so the difference is 19. The answer gives 2(9) + 1 = 19.

Answer: Answer: (2x2 + 1) square meters

In plain English

A polynomial is a sum of pieces. Each piece is a number times x with a whole-number power, like 3x2, −5x or 7.

Picture three kinds of tiles: big squares for x2, long bars for x, and small squares for 1. Adding two polynomials is pouring two piles of tiles together and counting each kind. Big squares only join big squares, bars join bars, and small squares join small squares.

Subtracting means taking away a whole pile. Everything in that pile goes, not just the first tile. That is why you flip the sign of every term in the second polynomial. If a tile in that pile is a negative one, taking it away adds to your total.

The answer is always another polynomial, because you only ever count tiles of each kind. To check your work, put the same number in for x in the problem and in your answer. Use a number like 3, not 0, 1 or 2.

See it

Vertical form: one column for each power of x x3 x2 x constant 3x3 0x2 5x −2 x3 −4x2 6x 8 first + second 4x3 −4x2 11x 6 sum 3 + 1 = 4 0 + (−4) = −4 5 + 6 = 11 −2 + 8 = 6 Read across: 4x3 − 4x2 + 11x + 6 The first polynomial has no x2 term, so its x2 column gets a placeholder 0.
Vertical form: line up like terms in columns, use a 0 placeholder for a missing power, and add straight down. (3x3 + 5x − 2) + (x3 − 4x2 + 6x + 8) = 4x3 − 4x2 + 11x + 6.
Subtracting flips every sign in the second polynomial (7x2 + 3x − 4) − (2x2 − 5x + 6) +2x2 −5x +6 −2x2 +5x −6 Multiply each term by −1 so every sign flips 7x2 + 3x − 4 − 2x2 + 5x − 6 = 5x2 + 8x − 10 Check x = 3: (68) − (9) = 59, and 5(9) + 24 − 10 = 59. Trap: flipping only the first sign gives 5x2 − 2x + 2, which is 41 at x = 3, not 59.
The minus sign multiplies every term of the second polynomial by −1, so every sign flips. (7x2 + 3x − 4) − (2x2 − 5x + 6) = 5x2 + 8x − 10.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Gaming

Coin tiers

A game pays out in three sizes of coin: gold coins worth 100 points, silver worth 10, and bronze worth 1. Haul A is 3 gold, 4 silver and 7 bronze, which is 347 points. Haul B is 2 gold, 8 silver and 1 bronze, which is 281 points. Together that is 5 gold, 12 silver and 8 bronze, which is 628 points.

How it connects: Gold, silver and bronze play the parts of x2, x and 1 when x = 10: (3x2 + 4x + 7) + (2x2 + 8x + 1) = 5x2 + 12x + 8. Like coins add with like coins, and at x = 10 the sum is 500 + 120 + 8 = 628, which matches 347 + 281. Limit: in the game you could trade 10 silver for 1 gold, but polynomials never trade up like that. The x stands for a number that could be anything, so 12x stays 12x.

Movie

The LEGO Movie

The LEGO Movie is set in a world built entirely out of LEGO bricks.

How it connects: Think of a polynomial as a model built from three kinds of bricks: x2 bricks, x bricks and single bricks. Snap two models together, or take some bricks out, and what you hold is still a model made from the same kinds of bricks. That is why adding or subtracting polynomials gives another polynomial. Limit: bricks of different kinds do not merge into one brick. Three x2 bricks and four x bricks stay as 3x2 + 4x. Also, some operations leave the set of models: 1 ÷ x (for x ≠ 0) is not a polynomial.

Idiom

The whole nine yards

To “go the whole nine yards” means to do something completely, all the way, with nothing left out.

How it connects: The minus sign in front of a polynomial goes the whole nine yards: it flips every term inside, not just the first one. −(x2 − 4x + 9) = −x2 + 4x − 9. Limit: the minus sign only flips signs. It never changes the sizes of the coefficients or exponents, and it reaches only the polynomial it sits in front of.

Sports

Box scores

A basketball box score has a column for each stat: points, rebounds and assists. To get a two-game total, you add column by column. Suppose in game 1 a player had 12 points, 5 rebounds and 0 assists, and in game 2 the player had 9 points, 3 rebounds and 4 assists. The totals are 21 points, 8 rebounds and 4 assists. Notice that the 0 assists in game 1 still gets written in its column.

How it connects: Vertical form works the same way. Write the two polynomials in rows, one column for each power of x, and add straight down each column. Points never merge with rebounds, and in the same way x2 terms, x terms and constants stay in separate columns. A missing power gets a 0 placeholder, just like the 0 assists in game 1. Limit: box-score columns hold different kinds of stats, but polynomial columns hold different powers of the same number x. The analogy is about keeping columns separate, not about what x means.

Proverb

A place for everything

The proverb “a place for everything and everything in its place” says a tidy space has a spot for every item, and every item is kept in its spot.

How it connects: Standard form gives every power its own place: x3 first, then x2, then x, then the constant. Vertical form gives every power its own column, even an empty one, which gets a 0. Limit: the order of terms never changes the value (3x + 2 and 2 + 3x are equal), so standard form is a habit that makes answers easy to compare, not a rule about what is true.

Food

Cancelling an order

A restaurant table’s bill is 40 dollars. One order on it had a sandwich (8 dollars), a drink (3 dollars) and a coupon line of −2 dollars, so that order came to 9 dollars. The server cancels that whole order, and the table’s bill drops to 31 dollars.

How it connects: 40 − (8 + 3 − 2) = 40 − 8 − 3 + 2 = 31. Cancelling the order removes every line in it, and removing the coupon line puts 2 dollars back on the bill, because subtracting −2 is the same as adding 2. That is the sign flip: −(8 + 3 − 2) = −8 − 3 + 2. Limit: with polynomials, the lines are terms, and a term only combines with terms of its own kind. A −2x term does not combine with a constant like −2.

Everyday joke

Back to school

Why did the polynomial go back to school? It wanted a higher degree.

How it connects: The degree of a polynomial in one variable is its highest exponent: 5x4 + x has degree 4. Limit: adding or subtracting can lower the degree instead of raising it, because leading terms can cancel: (5x4 + x) − (5x4 − 2) = x + 2, which has degree 1.

Memory help

Acronym

SIGN

Subtracting polynomials? Say SIGN. Spot the polynomial right after the minus sign. Invert the sign of every one of its terms. Group like terms and combine them. Number-check with x = 3 in the original and in your answer.

Rhyme

Match, then flip

Adding? Match kind with kind. Subtracting? Flip every sign you find in the second group, front to back. Then combine, and check to stay on track.

Picture it

The column ladder

Draw a table with one column for each power, from the highest down to the constant. Put each term in its column, with a 0 for any missing power. Add straight down each column, then read across. For subtraction, change every sign in the bottom row first, and then add.

Mnemonic

Big power first

Standard form is a staircase that steps down: the exponents get smaller from left to right, and the constant, if there is one, comes last. Say “biggest power first, constant last.” In 4 − x + 3x2, start with 3x2 and bring each sign along: 3x2 − x + 4.

Trap alert

Only the first term flips

(8x + 3) − (5x − 6) is not 3x − 3. The minus sign reaches every term in the second group, so −(5x − 6) = −5x + 6 and the answer is 3x + 9. If the minus sign does not touch every term of the second group, you are not finished.

Trap alert

Test values 0, 1 and 2 hide mistakes

At x = 0 only the constants are tested. At x = 1 every power of x equals 1. At x = 2, 2x2 and 4x are both 8. Use x = 3, but remember that no single value catches everything (at x = 3, x2 and 3x are both 9), so add a second value like 4 if you want more confidence.

Study notes

Key ideas

  • A polynomial is a monomial or a sum of monomials, with whole-number exponents. Monomial, binomial and trinomial have 1, 2 and 3 terms.
  • The degree of a polynomial is the highest degree among its terms (with one variable, that is the highest exponent; for a term with several variables, add their exponents). Standard form writes the terms from the highest degree to the lowest.
  • Adding polynomials means combining like terms: add the coefficients and keep the variable part.
  • A minus sign in front of a polynomial flips the sign of every term inside it. Subtracting is the same as adding the opposite.
  • The sum or difference of two polynomials is again a polynomial. Its degree can be lower than the degrees you started with.

Steps

  1. Write each polynomial in standard form (use a 0 placeholder for any missing power in vertical form).
  2. For subtraction, flip the sign of every term in the second polynomial and change the operation to addition.
  3. Group like terms and add their coefficients.
  4. Write the answer in standard form.
  5. Check with x = 3 in the original and in your answer.

Formulas

  • Adding: (ax2 + bx + c) + (dx2 + ex + f) = (a + d)x2 + (b + e)x + (c + f)
  • Subtracting: (ax2 + bx + c) − (dx2 + ex + f) = (a − d)x2 + (b − e)x + (c − f)
  • Minus sign in front: −(ax2 + bx + c) = −ax2 − bx − c
  • Subtracting is adding the opposite: A − B = A + (−B)
  • Degree of a sum or difference: At most the larger of the two degrees (it can be smaller if leading terms cancel)

Watch out

  • The minus sign flips every term of the second polynomial: −(3x − 8) = −3x + 8, not −3x − 8.
  • Adding like terms changes the coefficient, not the exponent: 5x3 + 2x3 = 7x3, not 7x6.
  • Unlike terms do not combine: 2x2 + 6x stays as it is.
  • The degree is the highest degree among the terms (with one variable, the highest exponent), not the degree of the first term written. Write the polynomial in standard form to see it.
  • Test with x = 3. The values 0, 1 and 2 can hide mistakes, and one matching value is a good sign, not a proof.

Why it works

  • Why can you combine like terms? 4x2 + 2x2 = (4 + 2)x2 is the distributive property read backward: 4 of something plus 2 of the same something is 6 of it.
  • Why does the minus sign flip every term? −(P) means (−1)(P), and the distributive property multiplies −1 by every term inside the parentheses.
  • Why is the result a polynomial? Adding or subtracting only changes coefficients, and the exponents stay whole numbers. That is what “closed under addition and subtraction” means.

Common mistakes

✗ Mistake(6x − 5) − (3x − 8) = 3x − 13 (only the first term of the second group is subtracted).
✓ InsteadThe minus sign applies to every term of the second group: −(3x − 8) = −3x + 8. So the answer is 6x − 5 − 3x + 8 = 3x + 3.
✗ Mistake5x3 + 2x3 = 7x6 (adding the exponents when adding).
✓ InsteadAdding like terms adds the coefficients and keeps the exponent: 5x3 + 2x3 = 7x3. Only multiplying adds exponents.
✗ Mistake2x2 + 6x = 8x3 (combining unlike terms).
✓ Insteadx2 and x are different variable parts, so they are not like terms. 2x2 + 6x is already as simple as it gets.
✗ MistakeThe degree of 3x + 8x5 is 1, because 3x comes first.
✓ InsteadWith one variable, the degree is the highest exponent, which is 5. In standard form, 8x5 + 3x, the highest power comes first.
✗ MistakeChecking with x = 0 or x = 1 is enough to prove an answer is right.
✓ InsteadAt x = 0 only the constants are tested, and at x = 1 every power of x equals 1, so both can hide mistakes. Use a value like 3. One match is a good sign, not a proof.
✗ MistakeAdding two polynomials of degree n always gives a polynomial of degree n.
✓ InsteadThe degree can drop when leading terms cancel: (x2 + 3x) + (−x2 + 4) = 3x + 4 has degree 1.

Teach it back

Explain to a friend how to subtract (8x2 − 3x + 2) − (3x2 + 5x − 6), why every sign in the second polynomial flips, and how to check your answer with x = 3.

Explaining it out loud shows what you really know. Did you…

  • I wrote −(3x2 + 5x − 6) = −3x2 − 5x + 6, flipping all three signs, and said the minus sign means multiplying by −1.
  • I combined like terms: 8x2 − 3x2 = 5x2, −3x − 5x = −8x and 2 + 6 = 8, to get 5x2 − 8x + 8.
  • I checked with x = 3: the original is 65 − 36 = 29 and the answer is 45 − 24 + 8 = 29.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Combining Like TermsThe Distributive PropertyMultiplying PolynomialsSpecial Products of BinomialsFactoring Out the Greatest Common FactorZero and Negative Exponents