➕ Algebra 1 · Exponents and Polynomials
Adding and Subtracting Polynomials: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.