# 4.4 Combining polynomials using addition and subtraction

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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: understand the concept of like terms, be able to combine like terms, be able to simplify expressions containing parentheses.

## Overview

• Like Terms
• Combining Like Terms
• Simplifying Expressions Containing Parentheses

## Like terms

Terms whose variable parts, including the exponents, are identical are called like terms . Like terms is an appropriate name since terms with identical variable parts and different numerical coefficients represent different amounts of the same quantity. As long as we are dealing with quantities of the same type we can combine them using addition and subtraction.

## Simplifying an algebraic expression

An algebraic expression can be simplified by combining like terms.

## Sample set a

Combine the like terms.

$6\text{\hspace{0.17em}}\text{houses}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\text{houses}=\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\text{houses}$ . 6 and 4 of the same type give 10 of that type.

$6\text{\hspace{0.17em}}\text{houses}+4\text{\hspace{0.17em}}\text{houses}+2\text{\hspace{0.17em}}\text{motels}=10\text{\hspace{0.17em}}\text{houses}+2\text{\hspace{0.17em}}\text{motels}$ . 6 and 4 of the same type give 10 of that type. Thus, we have 10 of one type and 2 of another type.

Suppose we let the letter $x$ represent "house." Then, $6x+4x=10x$ . 6 and 4 of the same type give 10 of that type.

Suppose we let $x$ represent "house" and $y$ represent "motel."

$6x+4x+2y=10x+2y$

## Practice set a

Like terms with the same numerical coefficient represent equal amounts of the same quantity.

Like terms with different numerical coefficients represent .

different amounts of the same quantity

## Combining like terms

Since like terms represent amounts of the same quantity, they may be combined, that is, like terms may be added together.

## Sample set b

Simplify each of the following polynomials by combining like terms.

$2x+5x+3x$ .
There are $2x\text{'}\text{s}$ , then 5 more, then 3 more. This makes a total of $10x\text{'}\text{s}$ .

$2x+5x+3x=10x$

$7x+8y-3x$ .
From $7x\text{'}\text{s}$ , we lose $3x\text{'}\text{s}$ . This makes $4x\text{'}\text{s}$ . The $8y\text{'}\text{s}$ represent a quantity different from the $x\text{'}\text{s}$ and therefore will not combine with them.

$7x+8y-3x=4x+8y$

$4{a}^{3}-2{a}^{2}+8{a}^{3}+{a}^{2}-2{a}^{3}$ .
$4{a}^{3},\text{\hspace{0.17em}}8{a}^{3},$ and $-2{a}^{3}$ represent quantities of the same type.

$4{a}^{3}+8{a}^{3}-2{a}^{3}=10{a}^{3}$

$-2{a}^{2}$ and ${a}^{2}$ represent quantities of the same type.

$-2{a}^{2}+{a}^{2}=-{a}^{2}$

Thus,

$4{a}^{3}-2{a}^{2}+8{a}^{3}+{a}^{2}-2{a}^{3}=10{a}^{3}-{a}^{2}$

## Practice set b

Simplify each of the following expressions.

$4y+7y$

$11y$

$3x+6x+11x$

$20x$

$5a+2b+4a-b-7b$

$9a-6b$

$10{x}^{3}-4{x}^{3}+3{x}^{2}-12{x}^{3}+5{x}^{2}+2x+{x}^{3}+8x$

$-5{x}^{3}+8{x}^{2}+10x$

$2{a}^{5}-{a}^{5}+1-4ab-9+9ab-2-3-{a}^{5}$

$5ab-13$

## Simplifying expressions containing parentheses

When parentheses occur in expressions, they must be removed before the expression can be simplified. Parentheses can be removed using the distributive property.

## Sample set c

Simplify each of the following expressions by using the distributive property and combining like terms.

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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
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Abigail
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NANO
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
how did you get the value of 2000N.What calculations are needed to arrive at it
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