In this chapter, you learn how to add and subtract polynomials. It begins with a discussion of the elementary concepts that you need to know to ensure your success when working with polynomials.
Terms and Monomials
In an algebraic expression, terms are the parts of the expression that are connected to the other parts by plus or minus symbols. If the algebraic expression has no plus or minus symbols, then the algebraic expression itself is a term.
Problem Identify the terms in the given expression.
a. 
b. 3x5
Solution
a. 
Step 1. The expression contains plus and minus symbols, so identify the quantities between the plus and minus symbols.
The terms are −8xy3, 
and 27.
b. 3x5
Step 1. There are no plus or minus symbols, so the expression is a term.
The term is 3x5.
A monomial is a special type of term that when simplified is a constant or a product of one or more variables raised to nonnegative integer powers, with or without an explicit coefficient.
Problem Specify whether the term is a monomial. Explain your answer.
a. −8xy3
b. 
c. 0
d. 3x5
e. 27
f. 4x−3y2
g. 
Solution
a. −8xy3
Step 1. Check whether −8xy3 meets the criteria for a monomial.
−8xy3 is a term that is a product of variables raised to positive integer powers, with an explicit coefficient of −8, so it is a monomial.
b. 
Step 1. Check whether 
meets the criteria for a monomial.
is a term, but it contains division by a variable, so it is not a monomial.
c. 0
Step 1. Check whether 0 meets the criteria for a monomial.
0 is a constant, so it is a monomial.
d. 3x5
Step 1. Check whether 3x5 meets the criteria for a monomial.
3x5 is a term that is a product of one variable raised to a positive integer power, with an explicit coefficient of 3, so it is a monomial.
e. 27
Step 1. Check whether 27 meets the criteria for a monomial.
27 is a constant, so it is a monomial.
f. 4x−3y2
Step 1. Check whether 4x−3y2 meets the criteria for a monomial.
4x−3y2 contains a negative exponent, so it is not a monomial.
g. 
Step 1. Check whether 
meets the criteria for a monomial.
contains a fractional exponent, so it is not a monomial.
Polynomials
A polynomial is a single monomial or a sum of monomials. A polynomial that has exactly one term is a monomial. A polynomial that has exactly two terms is a binomial. A polynomial that has exactly three terms is a trinomial. A polynomial that has more than three terms is just a general polynomial.
Problem State the most specific name for the given polynomial.
a. x2−1
b. 8a3 + 64b3
c. x2 + 4x−12
d. 
e. −2x5 + 5x4 −3x3 −7x2 + x + 4
Solution
a. x2−1
Step 1. Count the terms of the polynomial.
x2−1 has exactly two terms.
Step 2. State the specific name.
x2−1 is a binomial.
b. 8a3 + 64b3
Step 1. Count the terms of the polynomial.
8a3 + 64b3 has exactly two terms.
Step 2. State the specific name.
8a3 + 64b3 is a binomial.
c. x2 + 4x−12
Step 1. Count the terms of the polynomial.
x2 + 4x−12 has exactly three terms.
Step 2. State the specific name.
x2 + 4x−12 is a trinomial.
d. 
Step 1. Count the terms of the polynomial.
has exactly one term.
Step 2. State the specific name.
is a monomial.
e. −2x5 + 5x4 −3x3 −7x2 + x + 4
Step 1. Count the terms of the polynomial.
−2x5 + 5x4 −3x3 −7x2 + x + 4 has exactly six terms.
Step 2. State the specific name.
−2x5 + 5x4 −3x3 −7x2 + x + 4 is a polynomial.
Like Terms
Monomials that are constants or that have exactly the same variable factors (i.e., the same letters with the same corresponding exponents) are like terms. Like terms are the same except, perhaps, for their coefficients.
Problem State whether the given monomials are like terms. Explain your answer.
a. −10x and 25x
b. 4x2y3 and −7x3y2
c. 100 and 45
d. 25 and 25x
Solution
a. −10x and 25x
Step 1. Check whether −10x and 25x meet the criteria for like terms.
−10x and 25x are like terms because they are exactly the same except for their numerical coefficients.
b. 4x2y3 and −7x3y2
Step 1. Check whether 4x2y3 and −7x3y2 meet the criteria for like terms.
4x2y3 and −7x3y2 are not like terms because the corresponding exponents on x and y are not the same.
c. 100 and 45
Step 1. Check whether 100 and 45 meet the criteria for like terms.
100 and 45 are like terms because they are both constants.
d. 25 and 25x
Step 1. Check whether 25 and 25x meet the criteria for like terms.
25 and 25x are not like terms because they do not contain the same variable factors.
Finally, monomials that are not like terms are unlike terms.
Addition and Subtraction of Monomials
Because variables are standing in for real numbers, you can use the properties of real numbers to perform operations with polynomials.
Addition and Subtraction of Monomials
1. To add monomials that are like terms, add their numerical coefficients and use the sum as the coefficient of their common variable component.
2. To subtract monomials that are like terms, subtract their numerical coefficients and use the difference as the coefficient of their common variable component.
3. To add or subtract unlike terms, indicate the addition or subtraction.
Problem Simplify.
a. −10x + 25x
b. 4x2y3−7x3y2
c. 9x2 + 3x2 −7x2
d. 25 + 25x
e. 5x2 −7x2
Solution
a. −10x + 25x
Step 1. Check for like terms.
−10x + 25x
−10x and 25x are like terms.
Step 2. Add the numerical coefficients.
−10 + 25 = 15
Step 3. Use the sum as the coefficient of x.
−10x + 25x = 15x
b. 4x2y3−7x3y2
Step 1. Check for like terms.
4x2y3−7x3y2
4x2y3 and 7x3y2 are not like terms, so leave the problem as indicated subtraction: 4x2y3−7x3y2.
c. 9x2 + 3x2 −7x2
Step 1. Check for like terms.
9x2 + 3x2 −7x2
9x2, 3x2, and 7x2 are like terms.
Step 2. Combine the numerical coefficients.
9 + 3 − 7 = 5
Step 3. Use the result as the coefficient of x2.
9x2 + 3x2 −7x2 = 5x2
d. 25 + 25x
Step 1. Check for like terms.
25 + 25x
25 and 25x are not like terms, so leave the problem as indicated addition: 25 + 25x.
e. 5x2 −7x2
Step 1. Check for like terms.
5x2 −7x2
5x2and 7x2 are like terms.
Step 2. Subtract the numerical coefficients.
5−7 = −2
Step 3. Use the result as the coefficient of x2.
5x2 −7x2 = −2x2
Combining Like Terms
When you have an assortment of like terms in the same expression, systematically combine matching like terms in the expression. (For example, you might proceed from left to right.) To organize the process, use the properties of real numbers to rearrange the expression so that matching like terms are together (later, you might choose do this step mentally). If the expression includes unlike terms, just indicate the sums or differences of such terms. To avoid sign errors as you work, keep a − symbol with the number that follows it.
Problem Simplify 4x3 + 5x2 −10x + 25 + 2x3 −7x2 −5.
Solution
Step 1. Check for like terms.
4x3 + 5x2 −10x + 25 + 2x3 −7x2 −5
The like terms are 4x3and 2x3, 5x2and 7x2, and 25 and 5.
Step 2. Rearrange the expression so that like terms are together.
4x3 + 5x2 −10x + 25 + 2x3 −7x2 −5
= 4x3 + 2x3 + 5x2 −7x2 −10x + 25−5
Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.
= 6x3 + −2x2 −10x + 20
= 6x3 −2x2 −10x + 20
Step 4. Review your main result.
4x3 + 5x2 −10x + 25 + 2x3 −7x2 −5 = 6x3 −2x2 −10x + 20
Addition and Subtraction of Polynomials
Addition of Polynomials
To add two or more polynomials, add like monomial terms and simply indicate addition or subtraction of unlike terms.
Problem Perform the indicated addition.
a. (9x2 −6x + 2) + (−7x2 −5x + 3)
b. (4x3 + 3x2 −x + 8) + (8x3 + 2x−10)
Solution
a. (9x2 −6x + 2) + (−7x2 −5x + 3)
Step 1. Remove parentheses.
(9x2 −6x + 2) + (−7x2 −5x + 3)
= 9x2 −6x + 2−7x2 −5x + 3
Step 2. Rearrange the terms so that like terms are together. (You might do this step mentally.)
= 9x2 −7x2 −6x−5x + 2 + 3
Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.
= 2x2 −11x + 5
Step 4. Review the main steps.
(9x2 −6x + 2) + (−7x2 −5x + 3)
= 9x2 −6x + 2−7x2 −5x + 3 = 2x2 −11x + 5
b. (4x3 + 3x2 −x + 8) + (8x3 + 2x−10)
Step 1. Remove parentheses.
(4x3+ 3x2 −x + 8) + (8x3 + 2x−10)
= 4x3 + 3x2 −x + 8 + 8x3 + 2x−10
Step 2. Rearrange the terms so that like terms are together. (You might do this step mentally.)
= 4x3 + 8x3 + 3x2 −x + 2x + 8−10
Step 3. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.
= 12x3 + 3x2 + x−2
Step 4. Review the main steps.
(4x3 + 3x2 −x + 8) + (8x3 + 2x−10)
= 4x3 + 3x2 −x + 8 + 8x3 + 2x−10
= 12x3 + 3x2 + x−2
Subtraction of Polynomials
To subtract two polynomials, add the opposite of the second polynomial.
You can accomplish subtraction of polynomials by enclosing both polynomials in parentheses and then placing a minus symbol between them. Of course, make sure that the minus symbol precedes the polynomial that is being subtracted.
Problem Perform the indicated subtraction.
a. (9x2 −6x + 2)−(−7x2 −5x + 3)
b. (4x3 + 3x2 −x + 8)−(8x3 + 2x−10)
Solution
a. (9x2 −6x + 2)−(−7x2 −5x + 3)
Step 1. Remove parentheses.
(9x2 −6x + 2)−(−7x2 −5x + 3)
= 9x2 −6x + 2 + 7x2 + 5x−3
Step 2. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.
= 16x2 −x−1
Step 3. Review the main steps.
(9x2 −6x + 2)−(−7x2 −5x + 3)
= 9x2 −6x + 2 + 7x2 + 5x−3 = 16x2 −x−1
b. (4x3 + 3x2 −x + 8)−(8x3 + 2x−10)
Step 1. Remove parentheses.
(4x3 + 3x2 −x + 8)−(8x3 + 2x−10)
= 4x3 + 3x2 −x + 8−8x3 −2x + 10
Step 2. Systematically combine matching like terms and indicate addition or subtraction of unlike terms.
= −4x3 + 3x2 −3x + 18
Step 3. Review the main steps.
(4x3 + 3x2 −x + 8)−(8x3 + 2x−10)
= 4x3 + 3x2 −x + 8−8x3 −2x + 10
= 4x3 + 3x2 −3x + 18
Exercise 8
For 1–5, state the most specific name for the given polynomial.
1. x2 − x + 1
2. 125x3 − 64y3
3. 2x2 +7x − 4
5. 2x4 + 3x3 − 7x2 − x + 8
For 6–14, simplify.
6. − 15x +17x
7. 14xy3 − 7x3y2
8. 10x2 − 2x2 − 20x2
9. 10 +10x
10. 12x3 − 5x2 +10x − 60 +3x3 − 7x2 − 1
11. (10x2 − 5x + 3)+(6x2 +5x − 13)
12. (20x3 − 3x2 − 2x +5)+(9x3 + x2 + 2x − 15)
13. (10x2 − 5x +3)− (6x2 +5x − 13)
14. (20x3 − 3x2 − 2x + 5)− (9x3 + x2 +2x − 15)