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Multiplication of Polynomials: Rules and Examples, Summaries of Mathematics

Instructions on how to multiply a polynomial by a monomial and two polynomials using the distribution property and the rule for exponential expressions. It also introduces the foil method for multiplying two binomials and provides examples and exercises. From the math0301 course at the student learning assistance center of san antonio college.

What you will learn

  • How do you multiply a polynomial by a monomial?
  • What is the FOIL method and how is it used to multiply two binomials?
  • How do you multiply two polynomials?

Typology: Summaries

2021/2022

Uploaded on 09/12/2022

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Multiplication of Polynomials
Objective A To multiply a polynomial by a monomial
To multiply a polynomial by a monomial, use the Distribution Property and the Rule for
Multiplying Exponential Expressions.
Simplify: -3a(4a2 – 5a – 6) = -3a (4a2) – (-3a * 5a ) - (-3a * 6)
= -12a3 – (-15a2) – (-18a)
= -12a3 + 15a2 + 18a
Example 1
Simplify: (5x + 4)(-2x)
Solution: (5x + 4)(-2x) = 5x(-2x) + 4(-2x)
= –10x2 – 8x
Example 2
Simplify: 2a2b(4a2 – 2ab + b2)
Solution:
2a2b(4a2 – 2ab + b2) = 2a2b(4a2) – 2a2b(2ab) + 2a2b(b2)
= 8
a
4b – 4a3b2 + 2a2b3
Objective B To multiply two polynomials
Multiplication of two polynomials requires the repeated application of the Distributive
Property.
(y – 2)(y2 + 3y + 1) = (y – 2)(y2) + (y – 2)(3y) + (y – 2)(1)
= y3 – 2y2 + 3y2 – 6y + y - 2
= y3 + y2 – 5y - 2
A convenient method of multiplying two polynomials is to use a vertical format similar to
that used for multiplication of whole numbers.
Math0301
Student Learning Assistance Center - San Antonio College
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Multiplication of Polynomials

Objective A To multiply a polynomial by a monomial

To multiply a polynomial by a monomial, use the Distribution Property and the Rule for Multiplying Exponential Expressions.

Simplify: -3a(4a 2 – 5a – 6) = -3a (4a 2 ) – (-3a * 5a ) - (-3a * 6) = -12a 3 – (-15a 2 ) – (-18a) = -12a 3 + 15a^2 + 18a

Example 1

Simplify: (5x + 4)(-2x)

Solution: (5x + 4)(-2x) = 5x(-2x) + 4(-2x) = –10x^2 – 8x

Example 2

Simplify: 2a^2 b(4a 2 – 2ab + b 2 )

Solution: 2a 2 b(4a 2 – 2ab + b 2 ) = 2a 2 b(4a 2 ) – 2a 2 b(2ab) + 2a 2 b(b2^ ) = 8a 4 b – 4a 3 b2^ + 2a 2 b

Objective B To multiply two polynomials

Multiplication of two polynomials requires the repeated application of the Distributive Property.

(y – 2)(y 2 + 3y + 1) = (y – 2)(y 2 ) + (y – 2)(3y) + (y – 2)(1) = y 3 – 2y2^ + 3y 2 – 6y + y - 2 = y 3 + y 2 – 5y - 2

A convenient method of multiplying two polynomials is to use a vertical format similar to that used for multiplication of whole numbers.

Example of using the vertical format to perform the multiplication of polynomials

y2^ + 3y + y – 2 -2y 2 – 6y –2 [Multiply y 2 +3y + 1 by –2] y3^ + 3y 2 + y. [Multiply y 2 +3y + 1 by y] y3^ + y 2 - 5y – 2 [Add the terms in each column]

The resulting answer is the same under both formats.

Simplify: ( 2a^3 + a – 3 ) ( a + 5 )

2a 3 + 0a 2 + a – 3 a + 5 10a 3 + 0a 2 + 5a - 15 2a 4 + 0a 3 + a 2 – 3a 2a 4 +10a 3. + a 2 + 2a - 15

Objective C To multiply two binomials

It is frequently necessary to find the product of two binomials. The product can be found using a method called FOIL , which is based on the Distributive Property. The letters of FOIL stand for F irst, O utside, I nside, L ast.

Simplify: (2x + 3)(x + 5)

Multiply the First terms. (2x + 3)(x + 5) 2x * x = 2x 2

Multiply the Outer terms. (2x + 3)(x + 5) 2x * 5 = 10x

Multiply the Inner terms. (2x + 3)(x + 5) 3 * x = 3x

Multiply the Last terms. (2x + 3)(x + 5) 3 * 5 = 15

Add the combined terms.

F O I L (2x + 3)(x + 5) = 2x 2 + 10x + 3x + 15 = 2x 2 + 13x + 15

Objective E: To solve application problems

Example 3

The length of a rectangle is (x + 7) m. The width is (x – 4) m. Find the area of the rectangle in terms of variable x.

x – 4

x + 7

Strategy

To find the area, replace the variable L and W in the equation A = L * W by the given values and solve for A

Solution

A = L * W

A = (x + 7)(x –4) A = x 2 – 4x + 7x – A = x 2 + 3x –

The area is (x^2 + 3x – 28) m^2.