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Wize Grade 11 Mathematics Textbook > Polynomials and Operations

Multiplying Polynomials

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Dividing PolynomialsNext Section
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Multiplying Polynomials

To multiply two polynomials, every term in one must be multiplied by every term in the other polynomial.

Example 1

Multiply the following and simplify

(3x2−x+2)(2x−4)(3x^2 -x +2)(2x - 4)

(3x2−x+2)(2x−4)=6x3−12x2−2x2+4x+4x−8=6x3−14x2+8x−8\begin{aligned} (3x^2 - x + 2)(2x - 4) &= 6x^3 - 12x^2 - 2x^2 + 4x + 4x - 8 \\ &= 6x^3 - 14x^2 + 8x - 8 \end{aligned}


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Multiplying boxes

To better organize all of the terms that need to be multiplied, and later combine them, we can use a multiplying box.
  1. Write the terms of the polynomials along the sides of the box
  2. Multiply each column and row to fill in the boxes
  3. Combine similar terms
Example 2

Multiply the following and simplify

(3x2−x+2)(2x−4)(3x^2 - x + 2)(2x - 4)


=6x3−14x2+8x−8=6x^3 - 14x^2 + 8x - 8


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The FOIL Method

If both of the polynomials are binomials then you can use the FOIL method for multiplying. FOIL stands for
  • F - First terms
  • O - Outside terms
  • I - Inside terms
  • L - Last terms
Example 3

Multiply the following and simplify

(2x+5)(3x−2)(2x +5)(3x - 2)

  • F (2x)(3x)=6x2(2x)(3x) = 6x^2
  • O (2x)(−2)=−4x(2x)(-2) = -4x
  • I (5)(3x)=15x(5)(3x) = 15x
  • L (5)(−2)=−10(5)(-2) = -10
=6x2−4x+15x−10=6x2+11x−10\begin{aligned} &= 6x^2 - 4x + 15x - 10 \\ &= 6x^2 + 11x - 10 \end{aligned}
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Example: Multiplying Polynomials

At the National Gallery of Canada, a piece of art work is to be mounted on a solid steel backdrop.
This is done so that a steel boarder with a constant width will show around the art.


1. If the artwork measures 120cm by 85cm, create a function that describes the area of the steel backdrop in terms of the width of the border that will be seen x.

The shape of the steel backdrop is a rectangle, so we need to describe the width and the length.
To ensure that the width is the same on both sides, this will add 2x2x to the given dimensions of the art.

A(x)=(120+2x)(85+2x)A(x) = (120+2x)(85+2x)

2. Simplify the function by multiplying and combining any like terms

A(x)=(120+2x)(85+2x)=10200+410x+4x2\begin{aligned} A(x) &= (120 + 2x)(85 + 2x) \\ &= 10200 + 410x + 4x^2 \end{aligned}

Practice: Multiplying Polynomials

Use any method to multiply the following polynomials together.

1. (4x−2)(5x−7)(4x - 2)(5x - 7)

2. (6x−3)(6x3+3x−2)(6x - 3)(6x^3 + 3x - 2)

Practice: Multiply Polynomials

Use any method to multiply the expression.
(x−2)(x2+x)(3x−7)(x-2)(x^2 + x)(3x - 7)

Choose the answer that represents the simplified polynomial.


Practice: Multiplying Polynomials

The formula for the surface area of a rectangular solid is given by the formula: S=2ab+2bc+2acS=2ab + 2bc + 2ac

Find a function that describes the surface area in terms of aa given that
  • The length of bb is 3 more than aa
  • The length of cc is 1 less than aa