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Amy Ferguson Moncure

Algebra 1 · Unit 7

Multiply Polynomials

Multiply polynomials with at most five terms using distribute, FOIL, and area models.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Distribute

Every term in the first polynomial multiplies every term in the second. At most five terms before you combine.

Area model

A rectangle with sides (x + 3) and (x + 2) has area x² + 5x + 6. The four inner rectangles are the four products.

FOIL is distribute

First, Outer, Inner, Last is just a nickname for two binomials.

Exponents add

x × x² = x³. Multiply coefficients, add exponents of matching bases.

Level 1

Easy

A monomial times a binomial, or two small binomials.

Example
Multiply. 2x(x − 5)

2x(x) = 2x²

2x(−5) = −10x

2x² − 10x

Answer: 2x² − 10x

1
Multiply. 3(x + 4)
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3x + 12

Answer: 3x + 12

2
Multiply. x(x + 7)
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x² + 7x

Answer: x² + 7x

3
Multiply. (x + 2)(x + 3)
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x² + 3x + 2x + 6

x² + 5x + 6

Answer: x² + 5x + 6

4
An area model is a x-by-x square, an x-by-4 rectangle, a 3-by-x rectangle, and a 3-by-4. Write the product.
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(x + 3)(x + 4) = x² + 7x + 12

Answer: x² + 7x + 12

5
Multiply. 5x(2x − 1)
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10x² − 5x

Answer: 10x² − 5x

Level 2

Medium

Two binomials, or a square.

Example
Multiply. (x + 5)(x − 2)

x² − 2x + 5x − 10

x² + 3x − 10

Answer: x² + 3x − 10

1
Multiply. (2x + 3)(x + 4)
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2x² + 8x + 3x + 12

2x² + 11x + 12

Answer: 2x² + 11x + 12

2
Multiply. (x + 3)²
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(x + 3)(x + 3)

x² + 6x + 9

Answer: x² + 6x + 9

3
Multiply. (x − 4)(x − 4)
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x² − 8x + 16

Answer: x² − 8x + 16

4
A rectangle has sides (2x + 1) and (x + 6). Write the area as a trinomial.
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2x² + 12x + x + 6

2x² + 13x + 6

Answer: 2x² + 13x + 6

5
Multiply. (x + 2)(x² + 3x)
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x³ + 3x² + 2x² + 6x

x³ + 5x² + 6x

Answer: x³ + 5x² + 6x

Level 3

Hard

Up to five terms before combining, or a binomial times a trinomial.

Example
Multiply. (x + 2)(x² − 2x + 4)

x(x² − 2x + 4) = x³ − 2x² + 4x

2(x² − 2x + 4) = 2x² − 4x + 8

x³ − 2x² + 4x + 2x² − 4x + 8

x³ + 8

Answer: x³ + 8

1
Multiply. (2x − 1)(3x + 5)
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6x² + 10x − 3x − 5

6x² + 7x − 5

Answer: 6x² + 7x − 5

2
Multiply. (x + 1)(x + 2)(x + 3) by first multiplying two of them.
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(x + 1)(x + 2) = x² + 3x + 2

(x² + 3x + 2)(x + 3) = x³ + 3x² + 3x² + 9x + 2x + 6

x³ + 6x² + 11x + 6

Answer: x³ + 6x² + 11x + 6

3
An area model for (x + 5)(x − 2) has a negative 2-by-x rectangle. Write all four products, then the trinomial.
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x², −2x, 5x, −10

x² + 3x − 10

Answer: x² + 3x − 10

4
Multiply. (3x + 2)(3x − 2)
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9x² − 4

Difference of squares.

Answer: 9x² − 4

5
Multiply. x(x + 1)(x − 1)
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x(x² − 1) = x³ − x

Answer: x³ − x

Super Challenge

Super Challenge

1
Draw (in words) the area model for (x + a)(x + b) and write the four regions. Combine them.
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x-by-x square: x²

x-by-b and a-by-x rectangles: bx + ax = (a + b)x

a-by-b rectangle: ab

x² + (a + b)x + ab

Answer: x² + (a + b)x + ab

2
A student FOILs (2x + 3)(x + 4) as 2x² + 8 + 3x + 4. Name the missing products.
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Outer 2x(4) = 8x, not 8.

Last 3(4) = 12, not 4.

Correct: 2x² + 8x + 3x + 12 = 2x² + 11x + 12

Answer: 2x² + 11x + 12

3
Show two methods that (x + 3)² = x² + 6x + 9: distribute, and a square picture.
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Distribute: (x + 3)(x + 3) = x² + 3x + 3x + 9.

Picture: x² square, two 3-by-x rectangles, and a 9-unit square.

Answer: x² + 6x + 9

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