1. The height of an object launched t seconds is modeled by h(t) = -16t 2 + 32t + 25. Find the...
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Transcript of 1. The height of an object launched t seconds is modeled by h(t) = -16t 2 + 32t + 25. Find the...
1. The height of an object launched t seconds is modeled by h(t) = -16t2 + 32t + 25. Find the vertex and interpret what it means. What is the height of the object after 1.5 seconds?
2. The table below shows the average sale price p of a house in Suffolk County, Massachusetts for various years since 1988. Use your graphing calculator to find a quadratic model for this data. If this trend continues, what would the cost of a house be in 2010?
Algebra II 1
Factoring Quadratics
Algebra II
Greatest Common Factor
Trinomials with leading coefficient of 1 x2 + bx + c
Trinomials with leading coefficient other than 1 ax2 + bx + c
Difference of Two Squares
Four term polynomial – factor by grouping
Algebra II 3
Take out the greatest common factor of a trinomial by dividing each term by the GCF (greatest
common factor)Examples:1. 16x3 – 12x2 + 4x 2. 15xy2 – 25x2y GCF: 4x
4x(4x2 – 3x + 1)GCF: 5xy5xy(3y – 5x)
Algebra II 4
3. 27m3p2 + 9mp - 54p2 4. 10x – 40y
GCF: 9p9p(3m3p + m – 6p)
GCF: 1010(x – 4y)
Algebra II 5
x2 + bx + c = (x + )(x + )
The product of these numbers is c.
The sum of these numbers is b.
Algebra II 6
Algebra II 7
(x – 14)(x + 2)
(x + 5)(x – 2)
(x + 7)(x + 5)
(y – 12)(x + 2)
prime
(p + 8)(p – 5)
You should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial.
You can detect computational errors or errors in the signs of your numbers by checking your results.
8Algebra II
How are we going to factor if the leading coefficient is not
1?
The “X” Method ax2 + bx + c
9Algebra II
a c
b
the “#s” are factors of a cthat add up to
b
#1 #2
It is actually a graphic organization of “guess & check”
The “#s” are not what go in the binomials
Completely unnecessary if the leading coefficient is 1
10Algebra II
( )( )
Factor 8x2 – 14x + 5
Algebra II 11
4x– 5 2x – 1
40
-14
-4 -10
18
4215
( )( )
Factor 6x2 – 11x – 10
Algebra II 12
3x+2 2x – 5
-60
-11
-15 4
16
3211025
Factor 6x2 – 2x – 20 2(3x2 – x – 10)
2( )( )
Algebra II 13
3x+5 x – 2
-30
-1
-6 5
3111025
( )( )
Factor 21x2 – 13x + 2
Algebra II 14
3x– 1 7x – 2
42
-13
-6 -7
121
37
12
Factor 10a3 + 17a2 +3a a(10a2 + 17a + 3)
a( )
( )
Algebra II 15
2a+3 5a+1
30
17
2 15
110 13
25
( )( )
Factor 8x2 – x – 9
Algebra II 16
8x– 9 x + 1
-72
-1
8 -9
18
24
1933
Factor 4y2 – 2y – 12 2(2y2 – y – 6)
2( )( )
Algebra II 17
2y+3 y – 2
-12
-1
-4 3
21 1623
Factor 45a2 + 57a – 30 3(15a2 +19a – 10)
3( )( )
Algebra II 18
3a+5 5a – 2
-150
19
-6 25
115 1102535
( )( )
Factor 15x2 + 11x + 2
Algebra II 19
3x+1 5x+ 2
30
11
6 5
115
35
12
( )( )
Factor 15x2 – 29x – 2
Algebra II 20
15x + 1 x – 2
-30
-29
-30 1
115
35
12
(3x – 2)(x – 5)
(2x – 3)(2x + 1)
(7x – 1)(7x – 1)
prime
(5x + 7)(x + 2)
(3p – 5)(p + 2)
Algebra II 21
(x – 4)(8x + 3)
(3x + 1)(4x + 5)
prime
(2y + 1)(8y – 3)
(3x + 2)(3x + 2)
(2p – 1)(3p – 5)
Algebra II 22
Is it the difference of two Squares?
Factoring the Difference of Two Squares
a2 – b2 = (a + b)(a – b)
Algebra II 23
Factor: x2 – 9
(x – 3)(x + 3)
Algebra II 24
(2x – 3)(2x + 3)
(3x – 1)(3x + 1)
NOT A DIFF.
(1 – 5y)(1 + 5y)
(7y2 – 3z)(7y2 + 3z)
(9p – 5)(9p + 5)
Algebra II 25
prime
3(x2 – 9)3(x – 3)(x + 3)
4(x2 + x – 2)4(x – 1)(x + 2)
5(x2 – 4)5(x – 2)(x + 2)
2(7x2 + x – 6)2(7x – 6)(x + 1)
Algebra II 26
2u(u + 4)
2(5x2 + 17x + 14)2(5x + 7)(x + 2)
4x2(x2 – 16)4x2(x – 4)(x + 4)
3(10x2 – 19x + 7)3(2x – 1)(5x – 7)
Algebra II 27
If a polynomial has for terms: Factor the GCF out of the first two terms
Factor the GCF out of the second two terms
Factor out the common binomial and write as two binomials
28Algebra II
1.
29Algebra II