➕ Algebra 1 · Exponents and Polynomials
Multiplying 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.
The club mixer
Two clubs hold a mixer. The chess club sends 3 members and the robotics club sends 4. The rule of the night: every chess member has a short chat with every robotics member. That makes 3 × 4 = 12 chats.
How it connects: (a + b + c)(d + e + f + g) has 3 terms times 4 terms, so there are 3 × 4 = 12 products before you combine anything. Every term in the first factor meets every term in the second, and no pair meets twice. The limit: the chats only count the products. In algebra you also multiply each pair and add the results, and like terms can then merge, so the final answer often has fewer than 12 terms.
The sheet of brownies
A sheet of brownies is 12 inches by 8 inches. You make one straight cut that splits the 12-inch side into 10 inches and 2 inches, and one straight cut that splits the 8-inch side into 6 inches and 2 inches. That gives four pieces: 10 by 6, 10 by 2, 2 by 6 and 2 by 2 inches. Their areas are 60, 20, 12 and 4, and together they make 96 = 12 × 8 square inches.
How it connects: (10 + 2)(6 + 2) = 10 × 6 + 10 × 2 + 2 × 6 + 2 × 2 = 96. That is the box model: each cell is one product and the whole is the sum of the cells. Swap the 10 for x and the 6 for y and you get (x + 2)(y + 2) = xy + 2x + 2y + 4. The limit: a pan cannot have a negative side, so for something like (x − 3) the picture is only a guide. The box still works if you keep each term’s sign.
Better safe than sorry
This saying means it is wiser to take a small precaution now than to risk regret later. Packing an umbrella takes a second. Getting soaked can ruin the afternoon.
How it connects: A test value is your umbrella. Say you expand (x + 4)(x + 1) and get x2 + 4. At x = 3 the product is 7 × 4 = 28, but your answer gives 9 + 4 = 13, so something is missing. (The correct answer, x2 + 5x + 4, gives 9 + 15 + 4 = 28.) The limit: choose the value wisely. At x = 0 both sides would equal 4 and the mistake would slip through. And a match is a good sign, not a proof.
Tie up loose ends
To “tie up loose ends” means to finish the last small details of a job so that nothing is left hanging.
How it connects: After you multiply every term by every term, the loose ends are the like terms scattered around. In x2 + 5x + 3x + 15, the 5x and 3x are loose ends that tie together as 8x. The limit: only matching ends tie together. x2 and 15 stay as they are. And you tie up after all the products are written, not before.
A bigger practice court
A school replaces its small practice court with one that is exactly twice as long and twice as wide. If the old court was L feet long and W feet wide, its floor area was LW square feet. The new court is 2L feet by 2W feet.
How it connects: The new area is (2L)(2W) = (2 × 2)(L × W) = 4LW. The coefficients multiply (2 × 2 = 4), so the new floor needs 4 times the material, not 2 times. For a square court with side s, (2s)(2s) = 4s2: the coefficients multiply and the exponents add (s1 × s1 = s2). The limit: this works because both sides doubled. If only the length doubles, the area only doubles: (2L)(W) = 2LW.
Digging a room
In a block-building game you dig out a room that is x blocks long, x blocks wide and x blocks tall. The number of blocks you removed is x × x × x = x3. Then you make the room 2 blocks longer, so it is x + 2 blocks long, with the same width and height.
How it connects: The new volume is (x + 2)(x)(x) = x3 + 2x2 blocks. Each of the three sizes is a degree-1 factor, and the degrees add: 1 + 1 + 1 = 3. The limit: the degree only names the highest power. For a small room the other term matters a lot. At x = 3 the room holds 27 + 18 = 45 blocks, and the 2x2 part is 18 of them.
Not enough FOIL
Why couldn’t FOIL wrap up a binomial times a trinomial? It could only cover four of the six products, so two were left sticking out.
How it connects: FOIL names the four products of two binomials (2 × 2 = 4). A binomial times a trinomial has 2 × 3 = 6 products, so two would be left out. The rule underneath FOIL, every term times every term, works for any size. FOIL is not wrong for two binomials. It is just narrow.
Memory help
FOIL (two binomials only)
First, Outer, Inner, Last: the four products of two binomials. For (x + 2)(x + 7): First x2, Outer 7x, Inner 2x, Last 14. Add them and combine like terms: x2 + 9x + 14. FOIL lists the products of exactly two binomials. For a binomial times a trinomial there are six products, so use every term times every term.
PACE yourself
A four-step plan for any product. Pair every term in one factor with every term in the other and multiply. Add the products. Combine the like terms and write the answer in standard form. Evaluate a test value, such as x = 3, in the original and in your answer.
The multiplying rhyme
Multiply the numbers; for the same letter, exponents add. Skip a single product and your answer will be bad. Every term meets every term, count them one by one. Combine the like terms, and then you are done. Test with a three, not zero, one or two; those can hide mistakes from you.
Draw the box
Draw a table. Put one factor’s terms across the top and the other’s down the side, each with its sign. Fill every cell, color the like terms alike, and count the cells: the number of rows times the number of columns must equal the number of products you wrote.
The first-and-last trap
(x + 7)(x + 4) is not x2 + 28. That keeps only the First and Last products. The Outer and Inner products give 4x + 7x = 11x, so the answer is x2 + 11x + 28. At x = 3 the product is 10 × 7 = 70, but x2 + 28 gives only 37.
Multiply the numbers, add the exponents
5x3 × 2x4 is 10x7. The coefficients multiply (5 × 2 = 10, not 7) and the exponents add (3 + 4 = 7, not 12). Say it out loud: “numbers multiply, exponents add.” The exponents add only because both powers are of the same variable.