Copyright 2013, 2010, 2007, 2005, Pearson, Education, Inc.
5.2
Adding and
Subtracting
Polynomials
Polynomial Vocabulary
Term – a number or a product of a number and
variables raised to powers
Coefficient – numerical factor of a term
Constant – term which is only a number
Polynomial – a sum of terms involving variables
raised to a whole number exponent, with no
variables appearing in any denominator.
Polynomial Vocabulary
In the polynomial 7x5 + x2y2 – 4xy + 7
There are 4 terms: 7x5, x2y2, -4xy and 7.
The coefficient of term 7x5 is 7,
of term x2y2 is 1,
of term –4xy is –4 and
of term 7 is 7.
7 is a constant term.
Types of Polynomials
Monomial is a polynomial with one term.
Binomial is a polynomial with two terms.
Trinomial is a polynomial with three terms.
Degrees
Degree of a term
The degree of a term is the sum of the
exponents on the variables contained in the
term.
Degree of a constant is 0.
Degree of the term 5a4b3c is 8 (remember
that c can be written as c1).
Degrees
Degree of a polynomial
The degree of a polynomial is the greatest
degree of any term of the polynomial.
Degree of 9x3 – 4x2 + 7 is 3.
Evaluating Polynomials
Evaluating a polynomial for a particular value
involves replacing the value for the variable(s)
involved.
Example:
Find the value of 2x3 – 3x + 4 when x = 2.
2x3 – 3x + 4 = 2(2)3 – 3(2) + 4
= 2(8) + 6 + 4
= 6
Combining Like Terms
Like terms are terms that contain exactly the same
variables raised to exactly the same powers.
Warning!
Only like terms can be combined through addition and
subtraction.
Example:
Combine like terms to simplify.
x2y + xy – y + 10x2y – 2y + xy
= x2y + 10x2y + xy + xy – y – 2y
(Like terms are grouped
together)
= (1 + 10)x2y + (1 + 1)xy + (– 1 – 2)y = 11x2y + 2xy – 3y
Adding Polynomials
To Add Polynomials
To add polynomials, combine all like terms.
Example
Add: (3x – 8) + (4x2 – 3x + 3).
(3x – 8) + (4x2 – 3x + 3) = 3x – 8 + 4x2 – 3x + 3
= 4x2 + 3x – 3x – 8 + 3
= 4x2 – 5
Example
Add: 8 y  4 y  5 and 5 y  1 using a vertical
format.
3
2
2
8y  4y  5
3
2
5y  1
2
8 y3  y2  6
Subtracting Polynomials
To Subtract Polynomials
To subtract two polynomials, change the signs
of the terms of the polynomial being subtracted
and then added.
Example
Subtract 4 – (– y – 4).
4 – (– y – 4) = 4 + y + 4
= y+4+4
= y+8
Example
Subtract (– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7).
(– a2 + 1) – (a2 – 3) + (5a2 – 6a + 7)
= – a2 + 1 – a2 + 3 + 5a2 – 6a + 7
= – a2 – a2 + 5a2 – 6a + 1 + 3 + 7
= 3a2 – 6a + 11
Example
Subtract: (7x – 8) ‒ (3x – 12)
(7 x  8)  (3 x  12)  (7 x  8)  [ (3 x  12)]
 (7 x  8)  ( 3 x  12)
 7 x  8  3 x  12
 4x  4
Example
3
2
3
2
(2
x

8
x

7
x
)

(3
x

2
x
 3)
Subtract:
(2 x 3  8 x 2  7 x )  (3 x 3  2 x 2  3)
 (2 x  8 x  7 x )  ( 3 x  2 x  3)
3
2
3
2
 2 x  8 x  7 x  3x  2 x  3
3
2
3
  x  10 x  7 x  3
3
2
2
Example
Add or subtract:
(3x  7 xy  8 y )  ( 2 x  9 xy  y )
2
2
2
2
(3x 2  7 xy  8 y 2 )  ( 2 x 2  9 xy  y 2 )
 3x  7 xy  8 y  2 x  9 xy  y
2
 x  2 xy  7 y
2
2
2
2
2
Example
Subtract:
(9a b  7ab  4ab )  (6b a  3ab  4  10b )
2 2
2
2
2
 9a 2b2  7ab  4ab2  6b2a  3ab  4  10b 2
 9a b  4ab  4ab  10b  4
2 2
2
2