Multiplying Polynomials

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Multiplying Polynomials
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Lesson Objectives

• Know that a polynomial is a number, a variable,
  or a combination of a number and a variable.
• Know that a polynomial can be one or more terms.
• Know that monomials, binomials, and trinomials
  are all polynomials.
• Know how to multiply a polynomial by a monomial.

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Lesson Objectives

• Know how to multiply a binomial by a binomial.
• Know how to multiply a binomial by a trinomial.

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What is a Polynomial?

• A polynomial is a number, a variable, or a
  combination of a number and a variable.
• A polynomial can be one or more terms.
• Monomials, binomials, and trinomials are all
  polynomials.
• You have multiplied polynomials with one term
  when you multiplied 2x2 ? 3xy3.

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Multiplying a Polynomial by a Monomial

• A polynomial with one term is called a monomial.
• To multiply a polynomial with one term (monomial)
  by a polynomial with more than one term, use the
  distributive property.
• You multiply the term outside the parentheses by
  every term inside the parentheses.

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Multiplying a Polynomial by a Monomial

• Examples

2(a b 3)
(2 ? a)
(2 ? b)
(2 ? -3)
2a 2b -6
2a 2b ? 6
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Multiplying a Polynomial by a Monomial

• Examples

• 3x(x2 2x)

3x1 2
2 ? 3x1 1
3x3 6x2
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Multiplying a Binomial by a Binomial

• A binomial is a polynomial with two terms.
• To multiply a binomial by a binomial, you will
  use a method called FOIL.
• This process is called FOIL because you work
  the problem in this order

First
Outer
Inner
Last
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Multiplying a Binomial by a Binomial

• Example (x 2)(x 3)

• Multiply the first terms in each binomial.

(x 2)(x 3)
x2

• Multiply the two outer terms in each binomial.

(x 2)(x 3)
x2
3x

• Multiply the two inner terms in each binomial.

(x 2)(x 3)
x2 3x
2x

• Multiply the two last terms in each binomial.

(x 2)(x 3)
x2 3x 2x
6

• Simplify.

x2 5x 6
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Multiplying a Binomial by a Binomial

• Example (x 3)(x 1) (x 1)(x -1)

• Multiply the first terms in each binomial.

(x 3)(x -1)
x2

• Multiply the two outer terms in each binomial.

(x 3)(x -1)
x2
x

• Multiply the two inner terms in each binomial.

(x 3)(x -1)
x2 x
3x

• Multiply the two last terms in each binomial.

(x 3)(x -1)
x2 x 3x
3

• Simplify.

x2 2x 3
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Multiplying a Binomial by a Binomial

• Example (2x 1)(3x 2) (2x 1)(3x -2)

• Multiply the first terms in each binomial.

(2x 1)(3x -2)
6x2

• Multiply the two outer terms in each binomial.

(2x 1)(3x -2)
6x2
4x

• Multiply the two inner terms in each binomial.

(2x 1)(3x -2)
6x2 4x
3x

• Multiply the two last terms in each binomial.

(2x 1)(3x -2)
6x2 4x 3x
2

• Simplify.

6x2 x 2
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Multiplying a Binomial by a Trinomial

• A binomial has two terms, and a trinomial has
  three terms.

• Analyze how you would multiply a binomial by a
  trinomial.

• Example (x 2)(x2 2x 1)

• Multiply each term in the first polynomial with
  each in the second polynomial.

x(x2 2x 1)
x3 2x2 x
2(x2 2x 1)

• Multiply by 2 to each term in the second
  polynomial.

2x2 4x 2
2x2 4x 2
x3 2x2 x

• Simplify.

x3 4x2 5x 2
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Multiplying three Binomials

• Example (x 1)(x 1)(x 1)

• Multiply the first two factors. Then multiply the
  result by the third factor.

(x 1)(x 1)(x 1)

• Multiply (x 1)(x 1).

x2 x x 1

• Simplify.

x2 2x 1

• Then multiply (x2 2x 1)(x 1).

• Multiply x2(x 1).

x3 x2

• Multiply 2x(x 1).

2x2 2x

• Multiply 1(x 1).

x 1

• Simplify.

x3 x2 2x2 2x x 1
x3 3x2 3x 1