# Adding and Subtracting Polynomials – Tutorial and Practice

- Posted by Brian Stocker MA
- Date April 3, 2014
- Comments 1 comment

### Polynomials

Operations with Polynomial questions are on the CUNY, ACCUPLACER and PAX

BC Police – Ironworkers — IEBW — ACT — GED — COMPASS

See below for Tutorial and Practice questions

### Adding and subtracting operations with polynomials – a quick review

When we are adding or subtracting 2 or more polynomials, we have to first group the same variables (arguments) that have the same degrees and then add or subtract them. For example, if we have *ax ^{3}* in one polynomial (where

*a*is some real number), we have to group it with

*bx*from the other polynomial (where

^{3 }*b*is also some real number). Here is one example with adding polynomials:</p>

=(-x^{2}+2x+3)+(2x^{2}+4x-5)=

*-x ^{2}*

*+2x*

*+3*

*+2x*+4x

^{2}*-5*

*=*

*x ^{2}+6x-2*

We remove the brackets, and since we have a plus in front of every bracket, the signs in the polynomials don’t change.

We group variables with the same degrees: red is for second degree, and there we have *-1+2*, which is *1* and that’s how we got* x ^{2}*. Blue is for the first degree where we have

*2+4*which is

*6*, and the green is for the constants (real numbers) where we have

*3-5*which is

*-2*.

The principle is the same with subtracting, only we have to keep in mind that a minus in front of the polynomial changes all signs in that polynomial. Here is one example:

*(4x ^{3}-x^{2}+3)-(-3x^{2}-10)=*

*4x ^{3}-x^{2}+3+3x^{2}+10=*

*4x ^{3}+2x^{2}+13*

<p>We remove the brackets, and since wtation”>e have a minus in front of the second polynomial, all signs in that polynomial change. We have *-3x ^{2}*and with minus in front, it becomes a plus and same goes for

*-10*(red pluses).

Now we group the variables with same degrees: there is no variable with the third degree in the second polynomial, so we just write *4x ^{3}*. We group other variables the same way when we were adding polynomials.

### Adding and Subtracting Polynomials Practice Questions

**1. Add polynomials -3x^{2}+2x+6 and -x^{2}-x-1.**

a. *-2x ^{2}+x+5
*

*b.*

*-4x*

^{2}+x+5*c.*

*-2x*

^{2}+3x+5*d.*

*-4x*

^{2}+3x+5**2. Subtract polynomials 4x^{3}-2x^{2}-10 and 5x^{3}+x^{2}+x+5.**

a. *-x ^{3}-3x^{2}-x-15* b.

*9x*

c.

^{3}-3x^{2}-x-15*-x*

d.

^{3}-x^{2}+x-5*9x*

^{3}-x^{2}+x+5### Dividing Operations with Polynomials

**3. Divide x^{3}-3x^{2}+3x-1 by x-1.**

a. *x ^{2}-1
* b.

*x*

c.

^{2}+1*x*

d.

^{2}-2x+1*x*

^{2}+2x+1**4. Divide x^{2}-y^{2} by x-y.**

a. *x-y
* b.

*x+y*

c.

*xy*

d.

*y-x*

### Answer Key:

**1. B**

* -4x ^{2}+x+5
*

*(-3x*+ (

^{2}+2x+6)*-x*

^{2}-x-1)=*-3x*

^{2}+2x+6*-x*

^{2}-x-1=*-4x*

^{2}+x+5We remove the brackets and we group the variables by degrees.

** 2. A**

*-x ^{3}-3x^{2}-x-15*

*(4x*

^{3}-2x^{2}-10)-(5x^{3}+x^{2}+x+5)=*4x*

^{3}-2x^{2}-10-5x^{3}-x^{2}-x-5=*-x*

^{3}-3x^{2}-x-15We remove the brackets, but we change all signs in the second polynomial because of the minus. Now we group the variables by degrees.

**3. C**

*x ^{2}-2x+1
*

*(x*

^{3}-3x^{2}+3x-1) : (x-1)= x^{2}-2x+1*-(x*

^{3}-x^{2})*-2x*

^{2}+3x-1*-(-2x*

^{2}+2x)*x-1*

*-(x-1)*

*0*

**4. B**

*x+y*

* (x ^{2}-y^{2}) : (x-y) =x+y*

*-(x*

^{2}-xy)*xy-y*

^{2 }*-(xy-y*

^{2})*0*

**Written by**, Brian Stocker MA., Complete Test Preparation Inc.

**Date Published:**Thursday, April 3rd, 2014

**Date Modified:**Wednesday, July 21st, 2021

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## 1 Comment

nice. i will give you the feed back when am done testing my students.