<< Chapter < Page Chapter >> Page >
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 product of a monomial a and a binomial b plus c is equal to ab plus ac. This is the distributive property. In the expression, there are two arrows originating from the monomial, a, and pointing towards the terms b and c of the binomial.

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

10 a b 2 c ( 125 a 2 ) = 1250 a 3 b 2 c

Got questions? Get instant answers now!

Practice set a

Determine the following products.

( a 2 2 b + 6 ) 2 a

2 a 3 4 a b + 12 a

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 3 a 2 b ) ( 2 a b 2 + 4 b 3 )

6 a 3 b 3 + 12 a 2 b 4

Got questions? Get instant answers now!

5 m n ( m 2 n 2 + m + n 0 ) , n 0

5 m 3 n 3 + 5 m 2 n + 5 m n

Got questions? Get instant answers now!

Use a calculator. 6.03 ( 2.11 a 3 + 8.00 a 2 b )

12.7233 a 3 + 48.24 a 2 b

Got questions? Get instant answers now!

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.

Removal of a set of parentheses preceded by a plus sign using the distributive property. See the longdesc for a full description.

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

Removal of a set of parentheses preceded by a minus sign using the distributive property. See the longdesc for a full description.

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

  1. To remove a set of parentheses preceded by a " + " sign, simply remove the parentheses and leave the sign of each term the same.
  2. 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( 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

Got questions? Get instant answers now!

( 7 y 3 2 y 2 + 9 y + 1 ) = 7 y 3 + 2 y 2 9 y 1

Got questions? Get instant answers now!

Practice set b

Simplify by removing the parentheses.

( a 2 6 a + 10 )

a 2 6 a + 10

Got questions? Get instant answers now!

( x + 2 y )

x 2 y

Got questions? Get instant answers now!

( 5 m 2 n )

5 m + 2 n

Got questions? Get instant answers now!

( 3 s 2 7 s + 9 )

3 s 2 + 7 s 9

Got questions? Get instant answers now!

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,

Finding the product of the binomials 'a plus b' and 'c plus d', using the distributive property. See the longdesc for a full description.

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

a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
how did I we'll learn this
Noor Reply
f(x)= 2|x+5| find f(-6)
Prince Reply
f(n)= 2n + 1
Samantha Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask