4.5 Combining polynomials using multiplication

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to multiply a polynomial by a monomial, be able to simplify +(a + b) and -(a - b), be able to multiply a polynomial by a polynomial.

Overview

Multiplying a Polynomial by a Monomial
Simplifying $+ (a + b)$ and $- (a + b)$
Multiplying a Polynomial by a Polynomial

Multiplying a polynomial by a monomial

Multiplying a polynomial by a monomial is a direct application of the distributive property.

Distributive property

The distributive property suggests the following rule.

Multiplying a polynomial by a monomial

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.

Sample set a

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$10 a b^{2} c (125 a^{2}) = 1250 a^{3} b^{2} c$

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Practice set a

Determine the following products.

$3 (x + 8)$

$3 x + 24$

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$(2 + a) 4$

$4 a + 8$

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$(a^{2} - 2 b + 6) 2 a$

$2 a^{3} - 4 a b + 12 a$

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$8 a^{2} b^{3} (2 a + 7 b + 3)$

$16 a^{3} b^{3} + 56 a^{2} b^{4} + 24 a^{2} b^{3}$

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$4 x (2 x^{5} + 6 x^{4} - 8 x^{3} - x^{2} + 9 x - 11)$

$8 x^{6} + 24 x^{5} - 32 x^{4} - 4 x^{3} + 36 x^{2} - 44 x$

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$(3 a^{2} b) (2 a b^{2} + 4 b^{3})$

$6 a^{3} b^{3} + 12 a^{2} b^{4}$

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$5 m n (m^{2} n^{2} + m + n^{0}), n \neq 0$

$5 m^{3} n^{3} + 5 m^{2} n + 5 m n$

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$6.03 (2.11 a^{3} + 8.00 a^{2} b)$

$12.7233 a^{3} + 48.24 a^{2} b$

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Simplifying $+ (a + b)$ And $- (a + b)$

$+ (a + b)$ And $- (a + b)$

Oftentimes, we will encounter multiplications of the form

+ 1 (a + b)

or

- 1 (a + b)

These terms will actually appear as

+ (a + b)

and

- (a + b)

Using the distributive property, we can remove the parentheses.

The parentheses have been removed and the sign of each term has remained the same.

The parentheses have been removed and the sign of each term has been changed to its opposite.

To remove a set of parentheses preceded by a " $+$ " sign, simply remove the parentheses and leave the sign of each term the same.
To remove a set of parentheses preceded by a “ $-$ ” sign, remove the parentheses and change the sign of each term to its opposite sign.

Sample set b

Simplify the expressions.

$(6 x - 1)$ .

This set of parentheses is preceded by a “ $+$ ’’ sign (implied). We simply drop the parentheses.

$(6 x - 1) = 6 x - 1$

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$(14 a^{2} b^{3} - 6 a^{3} b^{2} + a b^{4}) = 14 a^{2} b^{3} - 6 a^{3} b^{2} + a b^{4}$

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$- (21 a^{2} + 7 a - 18)$ .

This set of parentheses is preceded by a “ $-$ ” sign. We can drop the parentheses as long as we change the sign of every term inside the parentheses to its opposite sign.

$- (21 a^{2} + 7 a - 18) = - 21 a^{2} - 7 a + 18$

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$- (7 y^{3} - 2 y^{2} + 9 y + 1) = - 7 y^{3} + 2 y^{2} - 9 y - 1$

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Practice set b

Simplify by removing the parentheses.

$(2 a + 3 b)$

$2 a + 3 b$

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$(a^{2} - 6 a + 10)$

$a^{2} - 6 a + 10$

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$- (x + 2 y)$

$- x - 2 y$

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$- (5 m - 2 n)$

$- 5 m + 2 n$

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$- (- 3 s^{2} - 7 s + 9)$

$3 s^{2} + 7 s - 9$

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Multiplying a polynomial by a polynomial

Since we can consider an expression enclosed within parentheses as a single quantity, we have, by the distributive property,

For convenience we will use the commutative property of addition to write this expression so that the first two terms contain $a$ and the second two contain $b$ .

Questions & Answers

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3

Esther Reply

derivative of logarithms function

Iqra Reply

how to solve this question

sidra

ex 2.1 question no 11

khansa

anyone can help me

khansa

question please

Rasul

ex 2.1 question no. 11

khansa

i cant type here

khansa

Find the derivative of g(x)=−3.

Abdullah Reply

any genius online ? I need help!!

Guzorochi Reply

how can i help you?

Pina

need to learn polynomial

Zakariya

i will teach...

nandu

I'm waiting

Zakariya

plz help me in question

Abish

How can I help you?

Tlou

evaluate the following computation (x³-8/x-2)

Murtala Reply

teach me how to solve the first law of calculus.

Uncle Reply

teach me also how to solve the first law of calculus

Bilson

what is differentiation

Ibrahim Reply

only god knows😂

abdulkadir

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)

Naufal Reply

fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5

diron

pweding paturo nsa calculus?

jimmy

how to use fundamental theorem to solve exponential

JULIA Reply

find the bounded area of the parabola y^2=4x and y=16x

Omar Reply

what is absolute value means?

Geo Reply

Chicken nuggets

Hugh

🐔

MM

🐔🦃 nuggets

MM

(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.

Ismael

find integration of loge x

Game Reply

find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3

Godwin Reply

how does this work

Brad Reply

Can calculus give the answers as same as other methods give in basic classes while solving the numericals?

Cosmos Reply

log tan (x/4+x/2)

Rohan

please answer

Rohan

y=(x^2 + 3x).(eipix)

Claudia

is this a answer

Ismael

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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