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Wize High School Grade 9 Math Textbook > Polynomial Expressions

Adding & Subtracting Polynomials

The Meaning of xx, x2x^2, x3x^3, and x\sqrt xPrevious Section
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Adding & Subtracting Polynomials

Recall Some Definitions

−3x2\Large{-3x^2}
  • This is a term
  • −3-3 is the coefficient (number part)
  • xx is the variable, which has an exponent of 2 (we call this "degree 2")

Two terms are called like terms if their variable parts are the exact same! (same variable, and same exponents)

Watch Out!
The coefficients (number parts) of like terms can be different.


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Adding and subtracting

We can add and subtract polynomials by grouping their like terms and adding or subtracting those coefficients.

Example 1
Simplify 2x+3+x2+5x−7−4x−3x22x+3+x^2+5x-7-4x-3x^2.

First rearrange so that all the like terms are next to each other:
=x2−3x2+2x+5x−4x+3−7=\colorTwo{x^2-3x^2}\colorThree{+2x+5x-4x}\colorFive{+3-7}

Then simplify:

=−2x2+3x−4=\colorTwo{-2x^2}\colorThree{+3x}\colorFive{-4}
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Example 2
Add the polynomials (5−xy+x2+3y)+(−4xy+x−y−y2+7)(5-xy+x^2+3y)+(-4xy+x-y-y^2+7).

When there is nothing or a + in front of a set of brackets, we can just remove the brackets:

5−xy+x2+3y−4xy+x−y−y2+75-xy+x^2+3y-4xy+x-y-y^2+7

First rearrange so that all the like terms are next to each other:

=x2−y2−xy−4xy+x+3y−y+5+7=\colorTwo{x^2}\colorFour{-y^2}\colorThree{-xy-4xy}\colorFive{+x}\colorOne{+3y-y}\red{+5+7}

Then simplify:

=x2−y2−5xy+x+2y+12=\colorTwo{x^2}\colorFour{-y^2}\colorThree{-5xy}\colorFive{+x}\colorOne{+2y}\red{+12}
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Example 3
Subtract the polynomials (x2y−2xy2+6+x2)−(3−x2y+5xy2−4x2)(x^2y-2xy^2+6+x^2)-(3-x^2y+5xy^2-4x^2).

Watch Out!
When you subtract a polynomial, all of the signs in that polynomial get flipped!

There is nothing in front of the first set of brackets, so we can just remove it:

=x2y−2xy2+6+x2−(+3−x2y+5xy2−4x2)=x^2y-2xy^2+6+x^2-(\bct +3\bct{-}x^2y\bct+5xy^2\bct-4x^2)

There is a - in front of the second set of brackets, so we need to flip all the signs in the second set of brackets:

=x2y−2xy2+6+x2−3+x2y−5xy2+4x2=x^2y-2xy^2+6+x^2\bct -3\bct+x^2y\bct-5xy^2\bct+4x^2

First rearrange so that all the like terms are next to each other:

=x2y+x2y−2xy2−5xy2+x2+4x2+6−3=\colorTwo{x^2y+x^2y}\colorThree{-2xy^2-5xy^2}\colorFive{+x^2+4x^2}\colorOne{+6-3}

Then simplify:

=2x2y−7xy2+5x2+3=\colorTwo{2x^2y}\colorThree{-7xy^2}\colorFive{+5x^2}\colorOne{+3}

Practice: Terms & Like Terms

Match the like-terms
A.
4xy24xy^2
B.
−3x2y2-3x^2y^2
C.
y2\dfrac{y}{2}
D.
−12xy-12xy
E.
xx
F.
5x2y5x^2y
3xy3xy
−4x2y-4x^2y
5xy25xy^2
x2y2x^2y^2
−x-x
yy

Practice: Adding and Subtracting Monomials

Simplify the following expressions.

a) −3x+7.5x-3x+7.5x

b) 12xy−13xy\dfrac{1}{2}xy-\dfrac{1}{3}xy

c) 4x3+23x34x^3+\dfrac{2}{3}x^3

d) −0.3t−1.52t-0.3t-1.52t
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Algebra Tiles

Some teachers will use algebra tiles to help students visualize what's actually happening when we add and subtract polynomials.
Opposite terms will have the exact same size but have different colours, they "cancel" out one another.

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Example
Simplify 2x+5+5x−7−4x2x+5+5x-7-4x.

We first represent this expression using algebra tiiles:

Now one by one, we can cancel out the opposite tiles:

We see that we only have this left over:

So the answer is 3x−23x-2.
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Example: Adding & Subtracting Polynomials

Simplify the following.

a) 3a+b−a−3+7b−5−10b3a+b-a-3+7b-5-10b

First rearrange so that all of the like terms are next to each other:

=3a−a+b+7b−10b−3−5=\colorTwo{3a-a}\colorThree{+b+7b-10b}\colorFive{-3-5}

Then simplify:

=2a−2b−8=\colorTwo{2a}\colorThree{-2b}\colorFive{-8}

So, the simplified answer is 2a−2b−8\boxed{2a-2b-8}.

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b) (2x2−5−3y+8x)−(7+5x2−9y+8x)(2x^2-5-3y+8x)-(7+5x^2-9y+8x)

Since there is nothing in front of the first set of brackets, we can just remove them:
=2x2−5−3y+8x−(7+5x2−9y+8x)=2x^2-5-3y+8x-(7+5x^2-9y+8x)

Since there is a - in front of the second set of brackets, we need to flip the signs in the second set of brackets:
=2x2−5−3y+8x−7−5x2+9y−8x=2x^2-5-3y+8x-7-5x^2+9y-8x

First rearrange so that all of the like terms are next to each other:
=2x2−5x2+8x−8x−3y+9y−5−7=\colorTwo{2x^2-5x^2}\colorThree{+8x-8x}\colorFive{-3y+9y}\colorOne{-5-7}

Then simplify:
=−3x2+0+6y−12=\colorTwo{-3x^2}\colorThree{+0}\colorFive{+6y}\colorOne{-12}

So, the simplified answer is −3x2+6y−12\boxed{-3x^2+6y-12}.

Practice: Adding & Subtracting Polynomials

Given the two polynomials (−7−3x2+6y−xy)(-7-3x^2+6y-xy) and (10−6x2−6y+5xy)(10-6x^2-6y+5xy) ,

a) add these two polynomials.

b) subtract these two polynomials.

Practice: Adding & Subtracting Polynomials

A local pet store charges a base fee of $10 for each dog visit and $15/hour that it takes to bathe the dog. They charge a base fee of $8 for each cat visit and $10/hour that it takes to bathe the cat.

Jess wants to bring her dog Appa and Cat Oreo into the store for a bath.

a) Represent the cost for bathing this dog as a polynomial.
b) Represent the cost for bathing this cat as a plynomial.
c) Represent the cost for bathing both pets as a polynomial.
d) If it takes 3 hours to bathe the dog Appa and 75 minutes to bathe the cat Oreo, calculate the combined cost for bathing both pets.


Practice: Adding & Subtracting Polynomials

Kevin added a mystery polynomial to 3x2+7x−y−5+y23x^2+7x-y-5+y^2, and he subtracted the same mystery polynomial from 3x2+7x−y−5+y23x^2+7x-y-5+y^2.

Unfortunately he was very clumsy and got coffee stains on the post-it note that he did his math on. Help him find this mystery polynomial.