Learn on PengiSaxon Algebra 1Chapter 6: Polynomials and Factoring

Lesson 58: Multiplying Polynomials

In this Grade 9 Saxon Algebra 1 lesson, students learn to multiply polynomials using the Distributive Property, the FOIL method (First, Outer, Inner, Last), and vertical multiplication. The lesson covers multiplying a monomial by a polynomial, multiplying two binomials, and finding the product of a binomial and a trinomial. Algebra tiles are also used to model binomial products visually before applying symbolic methods.

Section 1

📘 Multiplying Polynomials

New Concept

A second method of multiplying binomials is called the FOIL Method. FOIL stands for First, Outer, Inner, Last.

What’s next

Next, you'll apply the Distributive Property and the FOIL method to multiply various polynomials and solve problems involving area.

Section 2

Multiplying Monomials And Polynomials

Property

To find the product of a monomial and polynomial, multiply the monomial by each term in the polynomial. It's the distributive property in action! For example:

a(b+c)=ab+aca(b+c) = ab + ac

Explanation

Think of it like a newspaper delivery! The monomial is the delivery person who has to give a paper (multiply itself) to every single house (term) on the street (the polynomial). No house gets left out! Everyone gets a paper, making sure the distribution is complete and fair.

Examples

4x(x2+2y)=4x(x2)+4x(2y)=4x3+8xy4x(x^2 + 2y) = 4x(x^2) + 4x(2y) = 4x^3 + 8xy
−2ab(a2+b−c2)=−2ab(a2)+(−2ab)(b)+(−2ab)(−c2)=−2a3b−2ab2+2abc2-2ab(a^2 + b - c^2) = -2ab(a^2) + (-2ab)(b) + (-2ab)(-c^2) = -2a^3b - 2ab^2 + 2abc^2

Section 3

Example Card: Multiplying Binomials Using the Distributive Property

Let's break down multiplying two binomials using a property you already know well. This illustrates the key idea of using the Distributive Property.

Example Problem

Find the product of (x+8)(x+4)(x+8)(x+4).

Section 4

The FOIL Method

Property

FOIL stands for First, Outer, Inner, Last. It is a memory tool for multiplying two binomials by ensuring every term is distributed correctly.

(a+b)(c+d)=acFirst+adOuter+bcInner+bdLast(a+b)(c+d) = \underset{\text{First}}{ac} + \underset{\text{Outer}}{ad} + \underset{\text{Inner}}{bc} + \underset{\text{Last}}{bd}

Explanation

FOIL is your secret recipe for multiplying binomials! Just follow the steps in order: multiply the First terms, then the Outer terms, followed by the Inner terms, and finish with the Last terms. Add all these products together and simplify for your final answer.

Examples

(3x+2)(2x+4)=(3x)(2x)F+(3x)(4)O+(2)(2x)I+(2)(4)L=6x2+12x+4x+8=6x2+16x+8(3x+2)(2x+4) = \underset{\text{F}}{(3x)(2x)} + \underset{\text{O}}{(3x)(4)} + \underset{\text{I}}{(2)(2x)} + \underset{\text{L}}{(2)(4)} = 6x^2 + 12x + 4x + 8 = 6x^2 + 16x + 8
(8k−1)(−3k−5)=(8k)(−3k)F+(8k)(−5)O+(−1)(−3k)I+(−1)(−5)L=−24k2−40k+3k+5=−24k2−37k+5(8k-1)(-3k-5) = \underset{\text{F}}{(8k)(-3k)} + \underset{\text{O}}{(8k)(-5)} + \underset{\text{I}}{(-1)(-3k)} + \underset{\text{L}}{(-1)(-5)} = -24k^2 - 40k + 3k + 5 = -24k^2 - 37k + 5

Section 5

Example Card: Multiplying Binomials Using the FOIL Method

The FOIL method is a clever mnemonic for multiplying binomials. This example demonstrates this key idea.

Example Problem

Find the product of (3b+4)(2b+2)(3b+4)(2b+2).

Section 6

Multiplying A Binomial And A Trinomial

Property

To multiply a binomial and a trinomial, use the distributive property. Multiply each term in the binomial by the entire trinomial, then combine the resulting products.

(a+b)(c+d+e)=a(c+d+e)+b(c+d+e)(a+b)(c+d+e) = a(c+d+e) + b(c+d+e)

Explanation

This is just a bigger distribution party! Take the first term from the binomial and 'share' it with all three terms in the trinomial. Then, do the exact same thing with the second term from the binomial. Finally, add all the products together and simplify for your final answer.

Examples

(2x+3)(x2−5x−2)=2x(x2−5x−2)+3(x2−5x−2)=2x3−10x2−4x+3x2−15x−6=2x3−7x2−19x−6(2x+3)(x^2 - 5x - 2) = 2x(x^2 - 5x - 2) + 3(x^2 - 5x - 2) = 2x^3 - 10x^2 - 4x + 3x^2 - 15x - 6 = 2x^3 - 7x^2 - 19x - 6
(x−5)(3x2+2x−4)=x(3x2+2x−4)−5(3x2+2x−4)=3x3+2x2−4x−15x2−10x+20=3x3−13x2−14x+20(x-5)(3x^2+2x-4) = x(3x^2+2x-4) - 5(3x^2+2x-4) = 3x^3+2x^2-4x - 15x^2-10x+20 = 3x^3-13x^2-14x+20

Book overview

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Chapter 6: Polynomials and Factoring

  1. Lesson 1

    Lesson 51: Simplifying Rational Expressions with Like Denominators

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  2. Lesson 2

    Lesson 52: Determining the Equation of a Line Given Two Points

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  3. Lesson 3

    Lesson 53: Adding and Subtracting Polynomials

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  4. Lesson 4

    Lesson 54: Displaying Data in a Box-and-Whisker Plot

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  5. Lesson 5

    Lesson 55: Solving Systems of Linear Equations by Graphing

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  6. Lesson 6

    Lesson 56: Identifying, Writing, and Graphing Direct Variation

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  7. Lesson 7

    Lesson 57: Finding the Least Common Multiple

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  8. Lesson 8Current

    Lesson 58: Multiplying Polynomials

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  9. Lesson 9

    Lesson 59: Solving Systems of Linear Equations by Substitution

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  10. Lesson 10

    Lesson 60: Finding Special Products of Binomials

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Multiplying Polynomials

New Concept

A second method of multiplying binomials is called the FOIL Method. FOIL stands for First, Outer, Inner, Last.

What’s next

Next, you'll apply the Distributive Property and the FOIL method to multiply various polynomials and solve problems involving area.

Section 2

Multiplying Monomials And Polynomials

Property

To find the product of a monomial and polynomial, multiply the monomial by each term in the polynomial. It's the distributive property in action! For example:

a(b+c)=ab+aca(b+c) = ab + ac

Explanation

Think of it like a newspaper delivery! The monomial is the delivery person who has to give a paper (multiply itself) to every single house (term) on the street (the polynomial). No house gets left out! Everyone gets a paper, making sure the distribution is complete and fair.

Examples

4x(x2+2y)=4x(x2)+4x(2y)=4x3+8xy4x(x^2 + 2y) = 4x(x^2) + 4x(2y) = 4x^3 + 8xy
−2ab(a2+b−c2)=−2ab(a2)+(−2ab)(b)+(−2ab)(−c2)=−2a3b−2ab2+2abc2-2ab(a^2 + b - c^2) = -2ab(a^2) + (-2ab)(b) + (-2ab)(-c^2) = -2a^3b - 2ab^2 + 2abc^2

Section 3

Example Card: Multiplying Binomials Using the Distributive Property

Let's break down multiplying two binomials using a property you already know well. This illustrates the key idea of using the Distributive Property.

Example Problem

Find the product of (x+8)(x+4)(x+8)(x+4).

Section 4

The FOIL Method

Property

FOIL stands for First, Outer, Inner, Last. It is a memory tool for multiplying two binomials by ensuring every term is distributed correctly.

(a+b)(c+d)=acFirst+adOuter+bcInner+bdLast(a+b)(c+d) = \underset{\text{First}}{ac} + \underset{\text{Outer}}{ad} + \underset{\text{Inner}}{bc} + \underset{\text{Last}}{bd}

Explanation

FOIL is your secret recipe for multiplying binomials! Just follow the steps in order: multiply the First terms, then the Outer terms, followed by the Inner terms, and finish with the Last terms. Add all these products together and simplify for your final answer.

Examples

(3x+2)(2x+4)=(3x)(2x)F+(3x)(4)O+(2)(2x)I+(2)(4)L=6x2+12x+4x+8=6x2+16x+8(3x+2)(2x+4) = \underset{\text{F}}{(3x)(2x)} + \underset{\text{O}}{(3x)(4)} + \underset{\text{I}}{(2)(2x)} + \underset{\text{L}}{(2)(4)} = 6x^2 + 12x + 4x + 8 = 6x^2 + 16x + 8
(8k−1)(−3k−5)=(8k)(−3k)F+(8k)(−5)O+(−1)(−3k)I+(−1)(−5)L=−24k2−40k+3k+5=−24k2−37k+5(8k-1)(-3k-5) = \underset{\text{F}}{(8k)(-3k)} + \underset{\text{O}}{(8k)(-5)} + \underset{\text{I}}{(-1)(-3k)} + \underset{\text{L}}{(-1)(-5)} = -24k^2 - 40k + 3k + 5 = -24k^2 - 37k + 5

Section 5

Example Card: Multiplying Binomials Using the FOIL Method

The FOIL method is a clever mnemonic for multiplying binomials. This example demonstrates this key idea.

Example Problem

Find the product of (3b+4)(2b+2)(3b+4)(2b+2).

Section 6

Multiplying A Binomial And A Trinomial

Property

To multiply a binomial and a trinomial, use the distributive property. Multiply each term in the binomial by the entire trinomial, then combine the resulting products.

(a+b)(c+d+e)=a(c+d+e)+b(c+d+e)(a+b)(c+d+e) = a(c+d+e) + b(c+d+e)

Explanation

This is just a bigger distribution party! Take the first term from the binomial and 'share' it with all three terms in the trinomial. Then, do the exact same thing with the second term from the binomial. Finally, add all the products together and simplify for your final answer.

Examples

(2x+3)(x2−5x−2)=2x(x2−5x−2)+3(x2−5x−2)=2x3−10x2−4x+3x2−15x−6=2x3−7x2−19x−6(2x+3)(x^2 - 5x - 2) = 2x(x^2 - 5x - 2) + 3(x^2 - 5x - 2) = 2x^3 - 10x^2 - 4x + 3x^2 - 15x - 6 = 2x^3 - 7x^2 - 19x - 6
(x−5)(3x2+2x−4)=x(3x2+2x−4)−5(3x2+2x−4)=3x3+2x2−4x−15x2−10x+20=3x3−13x2−14x+20(x-5)(3x^2+2x-4) = x(3x^2+2x-4) - 5(3x^2+2x-4) = 3x^3+2x^2-4x - 15x^2-10x+20 = 3x^3-13x^2-14x+20

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Polynomials and Factoring

  1. Lesson 1

    Lesson 51: Simplifying Rational Expressions with Like Denominators

    LearnPractice
  2. Lesson 2

    Lesson 52: Determining the Equation of a Line Given Two Points

    LearnPractice
  3. Lesson 3

    Lesson 53: Adding and Subtracting Polynomials

    LearnPractice
  4. Lesson 4

    Lesson 54: Displaying Data in a Box-and-Whisker Plot

    LearnPractice
  5. Lesson 5

    Lesson 55: Solving Systems of Linear Equations by Graphing

    LearnPractice
  6. Lesson 6

    Lesson 56: Identifying, Writing, and Graphing Direct Variation

    LearnPractice
  7. Lesson 7

    Lesson 57: Finding the Least Common Multiple

    LearnPractice
  8. Lesson 8Current

    Lesson 58: Multiplying Polynomials

    LearnPractice
  9. Lesson 9

    Lesson 59: Solving Systems of Linear Equations by Substitution

    LearnPractice
  10. Lesson 10

    Lesson 60: Finding Special Products of Binomials

    LearnPractice