Transcript Chapter 5
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 5 u (1/3)m 2 -s 2 t 3
Binomial
A polynomial that has two terms 2x + 3 4x – 3y 3xy – 14 613 + 39z
Trinomial
A polynomial that has three terms 2x 2 – 3x + 1 14 + 32z – 3x mn – m 2 + n 2
Polynomial
Expressions with several terms that follow patterns.
4x 3 3b 2 + 3x 2 + 15x + 2 – 2b + 4
Coefficient
The constant (or numerical) factor in a monomial
3m 2 coefficient = 3
u coefficient = 1
-s 2 t 3 coefficient = -1
Like Terms
Terms that are identical or that differ only in their coefficients
Are 2x and 2y similar?
Are -3x 2 and 2x 2 similar?
Examples
x 2
x 2 + (-4)x + 5 – 4x + 5
What are the terms?
x 2 , -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?
2x 2
3x + 4x – 5
4x 2 – 5 + 4x + x 2 – x + 3x 2 – 5 + x 2
11-2 Multiply by a Monomial
Examples
(5a)(-3b)
3v 2 (v 2
12(a 2 + v + 1) + 3ab 2 – 3b 3 – 10)
11-3 Divide and Find Factors
GREATEST COMMON FACTOR
The greatest integer that is a factor of all the given integers.
2,3,5,7,11,13,17,19,23,29 Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.
GREATEST COMMON FACTOR
Find the GCF of 25 and 100 25 = 5 x 5 100 = 2 x 2 x 5 x 5 GCF = 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 =
Factoring Polynomials
vw + wx = w(v + x)
Factoring Polynomials
= 21x 2 – 35y 2
Factoring Polynomials
13e – 39ef =
Dividing Polynomials by Monomials
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 + 6x 2 2x
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 terms
O – multiply Outer terms
I – multiply Inner terms
L – multiply Last terms
Add all terms to get product
Example: (2a – b)(3a + 5b)
F – 2a · 3a
O – 2a · 5b
I – (-b) ▪ 3a
L - (-b) ▪ 5b
F – x
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O – x
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I – 6
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L – 6
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x 4 x 4 Example: (x + 6)(x +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 4x 3 + 4x 2 y 2 + xy + y 3 Group (4x 3 + 4x 2 y 2 ) and factor Group (xy + y 3 ) and factor 4x 2 (x +y 2 ) + y(x + y 2 ) Answer: (x +y 2 ) (4x 2 + y)
Example 2x 3 - 2x 2 y - 3xy 2 + 3y 3 + xz 2 – yz 2 Group (2x 3 - 2x 2 y 2 ) and factor Group (- 3xy 2 Group (xz 2 Answer: + 3y 3 ) and factor – yz 2 ) and factor
11-6 Special Factoring Patterns
11-6 Difference of Squares (a + b)(a – b)= a 2 - b 2 (x + 5) (x – 5) = x 2 - 25
11-6 Squares of Binomials (a + b) 2 = a 2 + 2ab + b 2 • (a - b) 2 = a 2 - 2ab + b 2 Also known as Perfect square trinomials
Examples (x + 3) 2 = ?
(y - 2) 2 = ?
(s + 6) 2 = ?
11-7 Factor Trinomials
Factoring Pattern for x 2 + bx + c, c positive x 2 + 8 x + 15 = (x + 3) (x + 5) Middle term is the sum of 3 and 5 Last term is the product of 3 and 5
Example y 2 + 14 y + 40 = (y + 10) (y + 4) Middle term is the sum of 10 and 4 Last term is the product of 10 and 4
Example y 2 – 11 y + 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 x 2 + bx + c, c negative x 2 - 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 y 2 + 6 y 40 = (y + 10) (y - 4) Middle term is the sum of 10 and -4 Last term is the product of 10 and - 4
Example y 2 – 7 y 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 ax 2 + 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 2x 2 + 7x – 9 List factors: (-2)(9) = -18 Factors: (-2)(9) add to 7 (2x 2 -2x) + (9x – 9) 2x(x -1) + 9(x – 1) (x-1)(2x +9)
Example 14x 2 - 17x + 5 List factors: (14)(5) = 70 Factors: (-7)(-10) add to -17 14x 2 (14x 2 -7x – 10x + 5 – 7x) + (-10x +5) 7x(2x-1)- 5(2x -1) (7x -5)(2x – 1)
Example 3x 2 - 11x - 4 List factors: (-12)(1) = -12 Factors: (-12)(1) add to -11 3x 2 (3x 2 -12x + 1x - 4 – 12x) + (1x -4) 3x(x-4) + 1(1x -4) (x -4)(3x + 1)
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