Download Your Notes - WordPress.com

Adding and Subtracting Polynomials Notes
Given Notes:
I. Adding and Subtracting Polynomials
A. Review of Vocabulary
1. Given the following example,
4x3+6x2+5x+8
a) 4x3, 6x2, 5x, and 8 are all terms
b) In the term 4x3, the number 4 is called
the coefficient of x3. Likewise, 6 is the
coefficient of x2, and 5 is the coefficient of x.
c) The term 8 in this expression is the
constant.
B. Review of Combining Like Terms
1. Like terms have exactly the same combination
of variables with the same exponents on the
variables. Only the coefficients may differ.
a) Examples of like terms:
(1) 19m5 and 14m5
(2) -37y9 and y9
b) Examples of unlike terms:
(1) 7x and 7y
(2) z4 and z
2. Like terms are combined by adding their
coefficients.
3. Examples of combining like terms:
9x6-14x6+x6
=(9-14+1)x6
=-4x6
all terms contain the variable x6
x6=1x6
12m2+5m+4m2 notice that all terms are not m2
=(12+4)m2+5m
=16m2+5m
3x4-3x2
these terms are unlike and cannot be
combined
4. Remember: Unlike terms have different
variables or different exponents on the same
variables.
B. Polynomials
1. Basic Definition
a) Polynomial basically means having
“many terms”
2. Specific types of polynomials
a) Monomial – a polynomial with only one
term
(1) Examples: 9m, -6y5, a2, and 6
Your Notes:
Adding and Subtracting Polynomials Notes
3. Specific types of polynomials (cont.)
a) Binomial – a polynomial with exactly
two terms
(1) Examples:
(a) -9x4+9x3
(b) 8m2+6m
(c) 3m5-9m2
b) Trinomial – a polynomial with exactly
three terms
(1) Examples:
(a) 9m3-4m2+6
(b) -3m5-9m2+2
c) An expression with more than three terms
is simply called a polynomial.
4. Degree of a Term
a) The degree of a term is the sum of the
exponents on the variables.
b) A constant term has a degree of 0.
c) Polynomials can only have whole
positive numbers as exponents (0,1,2,3…)
(1) Examples:
(a) 3x4 has degree 4
(b) 6y17 has degree 17
(c) 5x has degree 1
(i) 5x=5x1
(d) -7 has degree 0
(e) 2x2y has degree 3
(i) y has an exponent
of 1, so (2+1)=3
5. Degree of a Polynomial
a) The degree of a polynomial is the
greatest degree of any nonzero term of the
polynomial.
(1) Examples:
(a) 3x4-5x2+6 is of degree 4
(b) 5x+7 is of degree 1
(c) 3 is of degree 0
(d) x2y +xy-5xy2 is of degree 3
C. Adding Polynomials
1. To add two polynomials, add like terms.
a) Adding polynomials vertically
(1) (6x3-4x2+3)+(-2x3+7x2-5)
6x3 – 4x2 + 3
-2x3 + 7x2 - 5
4x3 + 3x2 - 2
write like terms in columns
(4x3+3x2-2)
final answer
now add, column by column
Your Notes:
Adding and Subtracting Polynomials Notes
Your Notes:
b) Adding polynomials vertically (cont.)
(1) (2x2-4x+3)+(x3+5x)
2x2 – 4x + 3
x3
+ 5x____
3
2
x + 2x + x + 3
write like terms in columns
leave spaces for missing terms
now add, column by column
(x3+2x2+x+3)
final answer
c) Adding polynomials horizontally
(1) (6x3-4x2+3)+(-2x3+7x2-5)
(6x3 - 4x2 + 3)+(-2x3 + 7x2 - 5) combine like terms
(6x3+(-2x3))+(-4x2+7x2)+(3+(-5))
(4x3+3x2-2)
final answer
(2) (3x2+4x+2)+(6x3-5x-7)
(3x2 + 4x + 2)+(6x3 - 5x - 7)
combine like terms
3
2
(6x )+(3x )+(4x+(-5x))+(2+(-7))
(6x3+3x2-x-5)
final answer
d) Hint: put negative numbers in
parentheses to keep from making sign
errors.
D. Subtracting Polynomials
1. To subtract two polynomials, change all the
signs of the second polynomial and add the result
to the first polynomial.
2. Remember: x-y is the same as x+(-y)
a) Subtracting Polynomials Vertically
(1) (14y3-6y2+2y-5)-(2y3-7y2-4y+6)
(14y3-6y2+2y-5) - (2y3-7y2-4y+6)
14y3 - 6y2 + 2y – 5
(-) 2y3 - 7y2 - 4y + 6
arrange like terms in columns
multiply the negative
(subtraction) sign through the
second row only. This will
change all signs on the row.
14y3 - 6y2 + 2y – 5
-2y3 + 7y2 + 4y - 6
12y3 + y2 +6y -11
now that the second row has
changed signs, add like terms.
(12y3+y2+6y-11)
final answer
Adding and Subtracting Polynomials Notes
b) Subtracting Polynomials Vertically (cont.)
(1) (4y3-16y2+2y)-(12y3-9y2+16)
4y3-16y2+2y
-12y3+9y2
-16
-8y3–7y2 +2y -16
arrange like terms in columns
change signs of the second row
(-8y3–7y2+2y-16)
final answer
c) Subtracting Polynomials Horizontally
(1) (5x-2)-(3x-8)
(5x-2) - (3x-8)
(5x-2) + (-3x+8)
(5x+(-3x)) + ((-2)+8)
(2x+6)
change the signs of the second
polynomial and add
final answer
(2) (11x3+2x2-8)-(6x3-4x2+2)
(11x3+2x2-8) - (6x3-4x2+2) change signs of second
(11x3+2x2-8) + (-6x3+4x2-2) polynomial and add
(11x3+(-6x3))+(2x2+4x2)+((-8)+(-2))
(5x3+6x2-10)
final answer
d) Remember:
Your Notes: