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Chapter 11
Polynomials
11-1
Add & Subtract Polynomials
Monomial
A constant, a variable, or a product of a constant and one or more variables
-7 5u (1/3)m2 -s2t3
Binomial
A polynomial that has two terms
2x + 3 4x – 3y3xy – 14 613 + 39z
Trinomial
A polynomial that has three terms
2x2 – 3x + 1 14 + 32z – 3xmn – m2 + n2
Polynomial
Expressions with several terms that follow patterns.
4x3 + 3x2 + 15x + 23b2 – 2b + 4
Coefficient
The constant (or numerical) factor in a monomial
3m2 coefficient = 3 u coefficient = 1 -s2t3 coefficient = -1
Like Terms
Terms that are identical or that differ only in their coefficients
Are 2x and 2y similar? Are -3x2 and 2x2 similar?
Examples
x2 + (-4)x + 5x2 – 4x + 5What are the terms?x2, -4x, and 5
Simplified Polynomial
A polynomial in which no two terms are similar.
The terms are usually arranged in order of decreasing degree of one of the variables
Are they Simplified?
2x2 – 5 + 4x + x2
3x + 4x – 54x2 – x + 3x2 – 5 + x2
11-2
Multiply by a Monomial
Examples
(5a)(-3b)3v2(v2 + v + 1)12(a2 + 3ab2 – 3b3 – 10)
11-3
Divide and Find Factors
The greatest integer that is a factor of all the given integers.
GREATEST COMMON FACTOR
Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.
2,3,5,7,11,13,17,19,23,29
Find the GCF of 25 and 100
25 = 5 x 5100 = 2 x 2 x 5 x 5GCF = 5 x 5 = 25
GREATEST COMMON FACTOR
Find the GCF of 12 and 36
12 = 36 =GCF =
GREATEST COMMON FACTOR
Find the GCF of 14,49 and 56
14 = 49 =56 =GCF =
GREATEST COMMON FACTOR
vw + wx = w(v + x)
Factoring Polynomials
21x2 – 35y2
=
Factoring Polynomials
13e – 39ef =
Factoring Polynomials
5m + 35 5
= 5(m+ 7)÷5= m + 7
Dividing Polynomials by Monomials
7x + 14 7
= 7x + 14 7 7 = x + 2
Dividing Polynomials by Monomials
6a + 8b 2
= 2(a +4b) ÷ 2 = a + 2b
Dividing Polynomials by Monomials
2x + 6x2
2x
Dividing Polynomials by Monomials
11-4
Multiply Two Binomials
Multiplying Binomials
When multiplying two binomials both terms of each binomial must be multiplied by the other two terms
Multiplying binomials
Using the F.O.I.L method helps you remember the steps when multiplying
F.O.I.L. Method
F – multiply First termsO – multiply Outer termsI – multiply Inner termsL – multiply Last termsAdd all terms to get product
Example: (2a – b)(3a + 5b)
F – 2a · 3aO – 2a · 5bI – (-b) ▪ 3aL - (-b) ▪ 5b
Example: (x + 6)(x +4)
F – x ▪ xO – x ▪ 4I – 6 ▪ xL – 6 ▪ 4
11-5
Find Binomial Factors in a Polynomial
Procedure
• Group the terms in the polynomial as pairs that share a common monomial factor
• Extract the monomial factor from each pair
Procedure
• If the binomials that remain for each pair are identical, write this as a binomial factor of the whole expression
• The monomials you extracted create a second polynomial. This is the paired factor for the original expression
Example
4x3 + 4x2y2 + xy + y3
Group (4x3 + 4x2y2) and factorGroup (xy + y3) and factor4x2(x +y2) + y(x + y2)Answer: (x +y2) (4x2 + y)
Example
2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2
Group (2x3 - 2x2y2 ) and factorGroup (- 3xy2 + 3y3) and factorGroup (xz2 – yz2) and factor Answer:
11-6
Special Factoring Patterns
11-6 Difference of Squares
(a + b)(a – b)= a2 - b2
(x + 5) (x – 5) = x2 - 25
11-6 Squares of Binomials
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
• Also known as Perfect square trinomials
Examples
(x + 3)2 = ?
(y - 2)2 = ?
(s + 6)2 = ?
11-7
Factor Trinomials
Factoring Pattern for x2 + bx + c, c positive
x2 + 8x + 15 = (x + 3) (x + 5)
Middle term is the sum of 3 and 5
Last term is the product of 3 and 5
Example
y2 + 14y + 40 = (y + 10) (y + 4)
Middle term is the sum of 10 and 4
Last term is the product of 10 and 4
Example
y2 – 11y + 18 = (y - 2) (y - 9)
Middle term is the sum of -2 and -9
Last term is the product of -2 and -9
Factoring Pattern for x2 + bx + c, c negative
x2 - x - 20 = (x + 4) (x - 5)
Middle term is the sum of 4 and -5
Last term is the product of 4 and - 5
Example
y2 + 6y - 40 = (y + 10) (y - 4)
Middle term is the sum of 10 and -4
Last term is the product of 10 and - 4
Example
y2 – 7y - 18 = (y + 2) (y - 9)
Middle term is the sum of 2 and -9
Last term is the product of 2 and -9
11-9
More on Factoring Trinomials
11-9 Factoring Pattern for ax2 + bx + c
• Multiply a(c) = ac• List the factors of ac• Identify the factors that add to b
• Rewrite problem and factor by grouping
Example 2x2 + 7x – 9
List factors: (-2)(9) = -18Factors: (-2)(9) add to 7(2x2 -2x) + (9x – 9)2x(x -1) + 9(x – 1)(x-1)(2x +9)
Example 14x2 - 17x + 5
List factors: (14)(5) = 70Factors: (-7)(-10) add to -1714x2 -7x – 10x + 5(14x2 – 7x) + (-10x +5)7x(2x-1)- 5(2x -1)(7x -5)(2x – 1)
Example 3x2 - 11x - 4
List factors: (-12)(1) = -12Factors: (-12)(1) add to -113x2 -12x + 1x - 4(3x2 – 12x) + (1x -4)3x(x-4) + 1(1x -4)(x -4)(3x + 1)
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