GED Mathematical Reasoning: Adding and Subtracting Polynomials

Adding polynomials

  • Step 1: Rewrite the problem without the parenthesis.
  • Step 2: Identify and combine “like terms.”

Remember: It is customary to begin with the term containing the largest exponent. And also – the sign before the term goes with the term. Meaning, if the sign is addition the term is positive. If the sign is subtraction, the term is negative.

 

Subtracting polynomials

  • Subtracting polynomials is similar to the process of adding polynomials, except for one additional step at the beginning of the process.
  • Before we combine like terms, we must first change the subtraction sign to addition and then change the sign of each term in the SECOND set of parenthesis to the opposite. Meaning, if the term is positive it will change to be negative. If the term is negative, it will change to be positive.
  • This goes back to the fact that subtraction means to “add the opposite.”

 

Example 1

Add: (3x^2 - 5x - 10) + (x^2 + 4x + 2)

 

In example one, we are being asked to add two polynomials. The parentheses are there to distinguish one polynomial from the other.

The first set of parenthesis contains:  3 “x” squared minus 5 “x” minus 10.

The second set of parenthesis contains:  “x” squared plus 4 “x” plus 2.

And we know to add these two polynomials together because of the addition sign in between the two sets of parenthesis.

We’ll first write the problem without parenthesis.

3x^2 - 5x - 10 + x^2 + 4x + 2

And then we’ll identify and combine the like terms. The like terms in example 1 are

3x^2 and x^2
-5x and 4x
-10 and 2

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3 plus 1 is 4, so 3 “x” squared combined with “x” squared is equal to 4 “x” squared.

3x^2 + x^2 = 3x^2 + 1x^2 = (3 + 1)x^2 = 4x^2

-5 “x” and 4 “x” add to equal -1 x, which is the same as simply negative “x”.

-5x + 4x = (-5 + 4)x = -1x = x

And -10 plus 2 is equal to -8.

-10 + 2 = -8

Putting these three terms together gives us a final answer of 4 “x” squared minus “x” minus 8.

(3x^2 - 5x - 10) + (x^2 + 4x + 2) = 4x^2 - x - 8

 

Example 2

Subtract: (7y^2 - 8y + 9) - (5y^2 - 25)

 

In example 2, we are being asked to subtract two polynomials.  Another way to look at things is to imagine that there’s -1 in front of the second set of parenthesis to distribute.

(7y^2 - 8y + 9) - (5y^2 - 25)
(7y^2 - 8y + 9) - 1(5y^2 - 25)

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Either way you look at it, you end up with something equivalent to:

(7y^2 - 8y + 9) + (-5y^2 + 25)

Notice that the subtraction sign changed to addition, the term positive 5 “y” squared became negative 5 “y” squared and the term negative 25 became positive 25. As you can see, now we have a problem whereby we are ADDING two polynomials. From here, we will rewrite the problem without parenthesis and then identify and combine like terms.

7y^2 - 8y + 9 +  -5y^2 + 25

The like terms are:

7y^2 and -5y^2
9 and 25

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7 “y” squared plus -5 “y” squared is equal to positive 2 “y” squared.

7y^2 +  -5y^2 = (7 +  -5)y^2 = 2y^2

-8 “y” has no other term to combine with so we’ll simply keep it as -8 “y”

-8y

And 9 plus 25 is equal to positive 34.

9 + 25 = 34

So the final answer is:  2 “y” squared minus 8 “y” plus 34.

Final answer: 2y^2 - 8y + 34

 

Example 3

Subtract: (5x - 11xy + y) - (9x - 11xy + 2y)

 

The processes for adding and subtracting polynomials are the same – no matter how many different variables are present.

Notice in example 3 that the two polynomials being subtracted contain two different variables – “x” and “y.” Just be careful that the terms you are combining in the last step are “like terms.” Especially when there are many variables and exponents floating around, it’s easy to make a small mistake that may lead to a wrong answer.

The first step when subtracting polynomials is to change the subtraction sign to addition and then change the sign of each term in the second set of parenthesis to the opposite.

(5x - 11xy + y) - (9x - 11xy + 2y)
 = (5x - 11xy + y) +(-9x + 11xy - 2y)

Our next step is to rewrite the polynomial without the parenthesis.

5x - 11xy + y - 9x + 11xy - 2y

Now we will identify and combine the like terms. The like terms are:

5x and -9x
-11xy and 11xy
y and -2y

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5x plus -9x is equal to -4x

5x +  -9x = (5 +  -9)x = -4x

When we combine -11xy and positive 11xy, the result is zero

-11xy + 11xy = (-11 + 11)xy = 0xy = 0

And y and -2y combine to be -1y or just –y.

y +  -2y = 1y +  -2y = (1 +  -2)y = -1y = -y

So the answer to example three is:  -4x  plus zero minus y. But we can omit the “zero” from the answer, giving us a final result of:  -4x minus y

Final Answer: -4x - y

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