Transcript Chapter 9

~ Chapter 9 ~
Polynomials and Factoring
Lesson 9-1 Adding & Subtracting Polynomials
Lesson 9-2 Mulitplying and Factoring
Lesson 9-3 Multiplying Binomials
Lesson 9-4 Multiplying Special Cases
Lesson 9-5 Factoring Trinomials of the Type x2 + bx + c
Lesson 9-6 Factoring Trinomials of the Type
ax2 + bx + c
Lesson 9-7 Factoring Special Cases
Lesson 9-8 Factoring by Grouping
Chapter Review
Adding & Subtracting Polynomials
Cumulative Review Chap 1-8
Adding & Subtracting Polynomials
Notes
Monomial – an expression that is a number, variable, or a product of a number
and one or more variables. (Ex. 8, b, -4mn2, t/3…)
(m/n is not a monomial because there is a variable in the denominator)
Degree of a Monomial
¾y
Degree: 1
3x4y2 Degree: 6
-8
Degree: 0
¾ y = ¾ y1… the exponent is 1.
The exponents are 4 and 2. Their sum is 6.
The degree of a nonzero constant is 0.
5x0 Degree = ?
Polynomial – a monomial or the sum or difference of two or more monomials.
Standard form of a Polynomial…
Simply means that the degrees of the polynomial terms decrease from left to
right.
5x4 + 3x2 – 6x + 3 Degree of each?
The degree of a polynomial is the same as the degree of the monomial with
the greatest exponent. What is the degree of the polynomial above?
Adding & Subtracting Polynomials
Notes
3x2 + 2x + 1
12
9x4 + 11x
5x5
The number of terms in a polynomial can be used to name the polynomial.
Classifying Polynomials
(1) Write the polynomial in standard form.
(2) Name the polynomial based on its degree
(3) Name the polynomial based on the number of terms
6x2 + 7 – 9x4
3y – 4 – y3
8 + 7v – 11v
Adding Polynomials
There are two methods for adding (& subtracting) polynomials…
Method 1 – Add vertically by lining up the like terms and adding the
coefficients.
Method 2 – Add horizontally by grouping like terms and then adding the
coefficients.
(12m2 + 4) + (8m2 + 5) =
Adding & Subtracting Polynomials
Notes
(9w3 + 8w2) + (7w3 + 4) =
Subtracting Polynomials
There are two methods for subtracting polynomials…
Method 1 – Subtract vertically by lining up the like terms and adding the
opposite of each term in the polynomial being subtracted.
Method 2 – Subtract horizontally by writing the opposite of each term in the
polynomial being subtracted and then grouping like terms.
(12m2 + 4) - (8m2 + 5) =
(30d3 – 29d2 – 3d) – (2d3 + d2)
Adding & Subtracting Polynomials
Homework
Homework – Practice 9-1
Multiplying & Factoring
Practice 9-1
Multiplying & Factoring
Practice 9-1
Multiplying & Factoring
Practice 9-1
Mulitplying & Factoring
Notes
Distributing a monomial
3x(2x - 3) = 3x(2x) – 3x(3) =
-2s(5s - 8) = -2s(5s) – (-2s) (8) =
Multiplying a Monomial and a Trinomial
4b(5b2 + b + 6) = 4b(5b2) + 4b(b) + 4b(6) =
-7h(3h2 – 8h – 1) =
2x(x2 – 6x + 5) =
Factoring a Monomial from a Polynomial
Find the GCF for 4x3 + 12x2 – 8x
4x3 = 2*2*x*x*x
12x2 = 2*2*3*x*x
8x = 2*2*2*x
What do they all have in common? 2*2*x = 4x
Multiplying & Factoring
Notes
Find the GCF of the terms of 5v5 + 10v3
Find the GCF of the terms of 4b3 – 2b2 – 6b
Factoring out a Monomial
Step 1: Find the GCF
Step 2: Factor out the GCF…
Factor 8x2 – 12x =
Factor 5d3 + 10d =
Factor 6m3 – 12m2 – 24m =
Factor 6p6 + 24p5 + 18p3 =
Multiplying & Factoring
Homework
Homework ~ Practice 9-2 even
Multiplying Binomials
Practice 9-2
Multiplying Binomials
Using the Distributive Property
Notes
Simplify (6h – 7)(2h + 3) = 6h(2h + 3) – 7(2h + 3) =
(5m + 2)(8m – 1) = 5m(8m – 1) + 2(8m - 1) =
(9a – 8)(7a + 4) = 9a(7a + 4) – 8(7a + 4) =
Multiplying using FOIL
F = First
O = Outer
I = Inner
L = Last
(6h – 7)(2h + 3) = 6h(2h) + 6h(3) + (-7)(2h) + (-7)(3)
12h2 + 18h + (-14h) + (-21) = 12h2 + 4h -21
(3x + 4)(2x + 5) =
(3x – 4)(2x – 5) =
Applying Multiplication of Polynomials
Determine the area of each rectangle and subtract the area of center
(x + 8)(x + 6) =
3x(x + 3) =
Multiplying Binomials
Notes
Multiplying a Trinomial and a Binomial
(2x – 3)(4x2 + x -6) = 2x(4x2) + 2x(x) + 2x(-6) -3(4x2) -3(x) -3(-6)
8x3 + 2x2 + (-12x) - 12x2 -3x + 18
Combine like terms = 8x3 – 10x2 – 15x + 18
You can also multiply using the vertical multiplication method…
Try this one…
(6n – 8)(2n2 + n + 7) =
Multiplying Binomials
Homework
Homework – Practice 9-3 even
Multiplying Special Cases
Practice 9-3
Multiplying Special Cases
Practice 9-3
Multiplying Special Cases
Notes
Finding the Square of a Binomial
(x + 8)2 = (x + 8)(x + 8) =
So…
(a + b)2 =
Rule: The Square of a Binomial
(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2
Find (t + 6)2
(5y + 1)2
(7m – 2p)2
Find the Area of the shaded region…
(x + 4)2 – (x – 1)2
Mental Math – Squares
312 = (30 + 1)2 = 302 + 2(30*1) + 12 = 900 + 60 + 1 = 961
Multiplying Special Cases
Notes
292 =
982 =
Difference of Squares
(a + b)(a – b) = a2 – ab + ab – b2
= a2 – b2
Find each product.
(d + 11)(d – 11) = d2 – 112 = d2 – 121
(c2 + 8)(c2 – 8) =
(9v3 + w4)(9v3 – w4) =
Mental Math
18 * 22 = (20 + 2)(20 – 2) = 202 – 22 = 400 – 4 = 396
59 * 61 =
87 * 93 =
Multiplying Special Cases
Homework
Homework – Practice 9-4
odd
Factoring Trinomials of the Type x2 + bx + c
Practice 9-4
Factoring Trinomials of the Type x2 + bx + c
Practice 9-4
Factoring Trinomials of the Type x2 + bx + c
Notes
Factoring Trinomials
x2 + bx + c
To factor this type of trinomial… you must find two numbers that
have a sum of b and a product of c.
Factor x2 + 7x + 12
Make a table…
Column 1 lists factors of c
12…
Column 2 lists the sum of those factors…
b
Row 3 – factors 3 & 4 with a sum of 7 fits so…
x2 + 7x + 12 = (x + 3)(x + 4)
Factor g2 + 7g + 10
Factor a2 + 13a + 30
Factoring Trinomials of the Type x2 + bx + c
Notes
Factoring x2 – bx + c
Since the middle term is negative, you must find the negative factors of c,
whose sum is –b.
Factor d2 – 17d + 42
> Make a table…
Row 3 – factors -3 & -14
with sum of -17
So… d2 – 17d + 42 = (d – 3)(d – 14)
Factor k2 – 10k + 25
Factor q2 – 15q + 36
Factoring Trinomials with a negative c (- c)
Factor m2 + 6m - 27
Make a table
Row 4 – factors 9 & -3 with sum of 6
Factoring Trinomials of the Type x2 + bx + c
Notes
So… m2 + 6m – 27 = (m + 9)(m – 3)
Factor p2 – 3p – 40
Factor m2 + 8m – 20
Factor y2 – y - 56
Factoring Trinomials of the Type x2 + bx + c
Homework
Homework ~ Practice 9-5 #1-30
Factoring Trinomials of the Type ax2 + bx + c
Practice 9-5
Factoring Trinomials of the Type ax2 + bx + c
Practice 9-5
Factoring Trinomials of the Type ax2 + bx + c
Practice 9-5
Factoring Trinomials of the Type ax2 + bx + c
Notes
Factoring Trinomials when c is positive
6n2 + 23n + 7…
Multiply a & c
So… 6n2 + 2n + 21n + 7
Factor using GCF
2n(3n + 1) + 7(3n + 1)
(2n + 7)(3n + 1) = 6n2 + 23n + 7
Try another one…
2y2 + 9y + 7
So… 2y2 + 2y +7y + 7
Factor…
2y(y + 1) + 7(y + 1)
(2y + 7)(y + 1)
What if b is negative?
6n2 - 2n – 21n + 7
6n2 – 23n + 7
Factors
of a*c
Sum
(=b)
6 and 7
13
3 and 14
17
2 and 21
23 √
Factoring Trinomials of the Type ax2 + bx + c
2n(3n - 1) – 7(3n – 1)
Notes
(2n – 7)(3n – 1)
Factors
of a*c
Sum
(=b)
Your turn… 2y2 – 5y + 2
7 and -8
-1
4 and -14
-10
Factoring Trinomials when c is negative…
2 and -28
-26 √
7x2 – 26x – 8
7x2 -28x + 2x – 8
7x(x – 4) + 2(x – 4)
(7x + 2)(x – 4)
Factor 5d2 – 14d – 3
5d2 -15d + 1d – 3
5d(d – 3) + 1(d – 3)
(5d + 1)(d - 3)
Factoring Trinomials of the Type ax2 + bx + c
Notes
Factoring Out a Monomial First
20x2 + 80x + 35
Factor out the GCF first…
5(4x2 + 16x + 7)… then factor 4x2 + 16x + 7
4x2 + 2x + 14x + 7
2x(2x + 1) + 7(2x + 1)
(2x + 7)(2x + 1) Remember to include the GCF in the final answer
5(2x + 7)(2x + 1)
Factor 18k2 – 12k - 6
6(3k2 – 2k – 1)
3k2 - 3k + 1k – 1
3k(k – 1) + 1(k – 1) = 6(3k + 1)(k - 1)
Factoring Trinomials of the Type ax2 + bx + c
Homework
Homework: Practice 9-6 first column
Factoring Special Cases
Practice 9-6
Factoring Special Cases
Practice 9-6
Factoring Special Cases
Notes
Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
So… x2 + 12x + 36 = (x + 6)2
And… x2 – 14x + 49 = (x – 7)2
What about… 4x2 + 12x + 9
Factoring a Perfect-Square Trinomial with a = 1 (ax2 + bx + c)
x2 + 8x + 16 =
n2 – 16n + 64 =
Factoring a Perfect-Square Trinomial with a ≠ 1
9g2 – 12g + 4
4t2 + 36t + 81
Factoring Special Cases
Notes
Factoring the Difference of Squares
a2 – b2 = (a + b)(a – b)
Or… x2 – 16 =
What about 25x2 – 81 =
Try x2 – 36
Factor 4w2 – 49
Look for common factors…
10c2 – 40 =
28k2 – 7 =
3c4 – 75 =
Factoring Special Cases
Homework
Homework: Practice 9-7 odd
#1-39
Factoring by Grouping
Practice 9-7
Factoring by Grouping
Practice 9-7
Factoring by Grouping
Practice 9-7
Factoring by Grouping
Notes
Factoring a Four-Term Polynomial
4n3 + 8n2 – 5n – 10
Factor the GCF out of each group of 2 terms.
? (4n3 + 8n2) - ? (5n + 10)
Factor 5t4 + 20t3 + 6t + 24
Before you factor, you may need to factor out the GCF.
12p4 + 10p3 -36p2 – 30p
Try… 45m4 – 9m3 + 30m2 – 6m (factor completely)
Finding the dimensions of a rectangular prism
The volume (lwh) of a rectangular prism is 80x3 + 224x2 + 60x. Factor
to find the possible expressions for the length, width, and height of the
prism.
Factoring by Grouping
Notes
Your turn…
Find expressions for possible dimensions of the rectangular prism…
V = 6g3 + 20g2 + 16g
V = 3m3 + 10m2 + 3m
Factoring by Grouping
Homework
Classwork – Practice 9-8 even
# 1-28
Factoring by Grouping
Practice 9-8
Chap 9 Quiz Review Lesson 7 & 8
Practice 9-8
~ Chapter 9 ~
Chapter Review
~ Chapter 9 ~
Chapter Review