Transcript Find the inverse of the function f(x)=3x-5.

The Distributive Property
(of * over +)
a(b+c)=ab+ac
Geometric Proof
b
a
ab
c
a
ac
Geometric Proof
a
b
c
ab
ac
a
Geometric Proof
b+c
a
ab+ac
a
Geometric Proof
b+c
a
a(b+c)
a
a(b+c)=ab+ac
Combine like terms
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2x+5x
2x+5x
(2+5)x
7x
ba+ca
(b+c)a
Don’t combine unlike terms
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2x2+5x
2x2+5x
(2x+5)x
stuck
ba+ca
(b+c)a
Distribute the negative
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2-(x-4)
2+-1(x+-4)
2+(-1)x+(-1)(-4)
2-x+4
6-x
turn - into +a(b+c)
ab+ac
The distributive property works
for any size sum
a(b+c+d+e+…)
=ab+ac+ad+ae+a…
Simplify completely:
(4 x  5 x  7)  (3 x  2 x  9)
2
2
a)12x2-3x-2
b)7x2-7x+16
c) 12x2-7x+16
d)7x2-3x+16
e)None of the above
(4x - 5x + 7) - (-3x + 2x - 9)
2
2
(4x - 5x + 7) + -1(-3x + 2x - 9)Distribute the negative
2
2
(4x 2 - 5x + 7) + (3x 2 - 2x + 9)
4x - 5x + 7 + 3x - 2x + 9
2
2
Combine like terms
7x - 7x +16
2
B
What most of you call FOILing
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(x-6)(x+2)
(x-6)(x+2)
a(b+c)
(x-6)x+(x-6)2 ab+ac
(x+-6)x+(x-6)2
(b+c)a
xx+-6x +(x-6)2
ba+ca
x2-6x +(x+-6)2
(b+c)a
x2-6x +2x+(2)(-6)
ab+ac
x2+-6x +2x-12
ba+ca
x2+(-6+2)x-12
(b+c)a
x2+-4x-12
Geometric “FOILing”
x
b
x
a
x2
ax
bx
ab
(x+a)(x+b)=x2+ax+bx+ab
Lattice Method
(2x3-3x2+2x+1)(-4x2+6)=?
2x3
-3x2
+2x
+1
-4x2
-8x5
+12x4
-8x3
-4x2
+6
12x3
-18x2
+12x
+6
=-8x5+12x4+4x3-22x2+12x+6
n
n
Find the product: (a + 6)(a - 6)
2n
a) a - 6
2
n
b) a - 6
2
n
c) a - 36
2
n
d) a + 36
e) None of the above
n
n
Find the product: (a + 6)(a - 6)
Remember rules of exponents! anan=an+n=a2n
an
+6
an
a2n
6an
-6
-6an
-36
2n
a - 36
E) None of
the above
Factor and Factoring
• A factor is something being multiplied
• 2 and x are factors of 2x
• Factoring is the art of turning something
into a multiplication.
• Ex: 6 can be factored into 2*3:
• 2 and 3 are factors of 6.
• Factoring nicely can be hard because you
have to add information.
Factoring Trinomials pt1
• (x+a)(x+b)
• x2+(a+b)x+ab
• x2-5x+6 =
(x+a)(x+b)
• what are a and b?
Factoring Trinomials pt1
• (x+a)(x+b)
• x2+(a+b)x+ab
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x2+-5x+6 = (x+a)(x+b)
what are a and b?
a+b=-5 ab=6
a=-2, b=-3
(x-2)(x-3)
Factoring Trinomials pt2
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(cx+a)(dx+b)
cdx2+(da+cb)x+ab
2x2+7x-15 = (cx+a)(dx+b)
what are a, b, c and d?
Factoring Trinomials pt2
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(mx+a)(nx+b)
mnx2+(na+mb)x+ab
2x2+7x-15 = (mx+a)(nx+b)
what are a, b, m and n?
mn=2
na+mb=7 ab=-15
m=2, n=1, a=-5, b=3?
(1)(-5)+2(3)=1 
m=2, n=1, a=5, b=-3?
(1)(5)+2(-3)=-1 
m=2, n=1, a=3, b=-5?
(1)(3)+2(-5)=-7 
m=2, n=1, a=-3, b=5?
(1)(-3)+2(5)=7 
(2x-3)(x+5)
Factoring By Grouping
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3x3-6x2+9x-18
(3x3-6x2)+(9x-18)
3x2(x-2)+9(x-2)
(3x2+9)(x-2)
The trick is to pull something out of each
group so that what is left over is the same.
Factoring By Grouping
• The trick is to pull something out of each
group so that what is left over is the same.
• 4x3-6x2-10x+15
• 4x3-6x2+-10x+15
• (4x3-6x2)+(-10x+15)
• 2x2(2x-3)+5(-2x+3)
Not the same!
• 2x2(2x-3)+-5(2x-3)
The same!
• (2x2-5)(2x+3)
Factor:
3 x  7 x  20
2
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b)
c)
d)
e)
(x-5)(x+4)
(3x+5)(x-4)
(3x-5)(x+4)
(x-4)(3x+5)
None of the above
3 x  7 x  20
2
3x2+7x-20
(mx+a)(nx+b)= mnx2+(na+mb)x+ab
mn=3
m=3, n=1
ab=-20 a=2, b=-10; a=-2, b=10;
a=-10, b=2; a=10, b=-2;
a=4, b=-5; a=-4, b=5
a=-5, b=4; a=5, b=-4
…
1a+3b=7
1(-5)+3(4)=-5+12=7
a=-5, b=4
c) (3x-5)(x+4)