AA Section 11-3 Day 2

Section 11-3The fun part

Tuesday, March 3, 2009

Discriminant Theorem for Factoring Quadratics



A polynomial ax2 + bx + c can be factored into first degree (linear) polynomials IFF the discriminant D = b2 - 4ac is a perfect square.



A polynomial ax2 + bx + c can be factored into first degree (linear) polynomials IFF the discriminant D = b2 - 4ac is a perfect square.

Is there anything the discriminant CAN’T do?!


Prime/Irreducible


Prime/Irreducible

A polynomial that cannot be factored into lower-degree polynomials with rational coefficients.


Example 1Determine whether the following are prime.

a. x2 - 9 b. x2 - 10



a. x2 - 9 b. x2 - 10

b2 - 4ac



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36

36 is a perfect square



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36


This one is factorable!



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36



It’s also a difference of squares!



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36




b2 - 4ac



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36




b2 - 4ac

= 02 - 4(1)(-10)



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36




b2 - 4ac

= 02 - 4(1)(-10)

= 40



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36




b2 - 4ac

= 02 - 4(1)(-10)

= 40

40 is not a perfect square



a. x2 - 9 b. x2 - 10

b2 - 4ac

= 02 - 4(1)(-9)

= 36




b2 - 4ac

= 02 - 4(1)(-10)

= 40

40 is not a perfect square

This one is not factorable!


Example 2Can 6m2 - 7m - 20 be factored? If so, factor it.



b2 - 4ac



b2 - 4ac = (-7)2 - 4(6)(-20)



b2 - 4ac = (-7)2 - 4(6)(-20) = 49



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529




b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529

529 is a perfect square (23 ⋅ 23 = 529)



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529

529 is a perfect square (23 ⋅ 23 = 529)so this quadratic is factorable.



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529


. . .



b2 - 4ac = (-7)2 - 4(6)(-20) = 49 + 480 = 529


. . .

How do we factor this?


There is a pattern!First, multiply a and c.



(6)(-20) =



(6)(-20) = -120



(6)(-20) = -120Now, factor this product into two numbers that add up to b. Since b is negative, the larger factor must also be negative.




(2)(-60) = -120




(2)(-60) = -120 2 - 60 =




(2)(-60) = -120 2 - 60 = -58




(2)(-60) = -120 2 - 60 = -58≠ -7




(2)(-60) = -120 2 - 60 = -58≠ -7

The two factors need to be much closer.




(2)(-60) = -120 2 - 60 = -58≠ -7


(8)(-15) = -120




(2)(-60) = -120 2 - 60 = -58≠ -7


(8)(-15) = -120 8 - 15 =




(2)(-60) = -120 2 - 60 = -58≠ -7


(8)(-15) = -120 8 - 15 = -7




(2)(-60) = -120 2 - 60 = -58≠ -7


(8)(-15) = -120 8 - 15 = -7

GOOD NEWS!!!


We’re going to take our quadratic and rewrite the middle term.

6m2 - 7m - 20



6m2 - 7m - 20

6m2 + 8m - 15m - 20



6m2 - 7m - 20

6m2 + 8m - 15m - 20Group the first two and last two terms (including minus signs).



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m(3m



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m(3m + 4)



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m(3m + 4) - 5



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m(3m + 4) - 5(3m



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)

Factor out the GCF of each binomial.2m(3m + 4) - 5(3m+ 4)



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)


You will notice that the “stuff” inside the parentheses is the same. This is one of our factors. The “stuff” that’s left over

makes our other factor!



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)




(3m + 4)



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)




(3m + 4)(2m - 5)



6m2 - 7m - 20


(6m2 + 8m) + (- 15m - 20)




(3m + 4)(2m - 5)

And we have our answer!Tuesday, March 3, 2009

Process for factoring ax2 + bx + c



1. Multiply a and c. Factor this number so that the two factors add up to b. (The larger factor will take the sign of b.)




2. Group the first two and last two terms (including negative signs) and then factor out the GCF from each binomial.




2. Group the first two and last two terms (including negative signs) and then factor out the GCF from each binomial.

3. Rewrite as the two factors: The “stuff” inside the parentheses is one, and the “stuff” outside is the other.


Example 3: Factor.12x3 - 28x2 - 24x



What do you do first?



What do you do first?4x



What do you do first?4x (3x2



What do you do first?4x (3x2 - 7x



What do you do first?4x (3x2 - 7x - 6)




Is the quadratic factorable?




Is the quadratic factorable?b2 - 4ac




Is the quadratic factorable?b2 - 4ac = (-7)2 - 4(3)(-6)




Is the quadratic factorable?b2 - 4ac = (-7)2 - 4(3)(-6) = 121





Okay, let’s do it!






(3)(-6)






(3)(-6) = -18






(3)(-6) = -18 2 - 9 = -7






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)+ 2






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)+ 2(x - 3)]






(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)+ 2(x - 3)]

4x[Tuesday, March 3, 2009





(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)+ 2(x - 3)]

4x[(x - 3)Tuesday, March 3, 2009





(3)(-6) = -18 2 - 9 = -74x(3x2 - 7x - 6)

4x(3x2 - 9x + 2x - 6)4x[(3x2 - 9x) + (2x - 6)]4x[3x(x - 3)+ 2(x - 3)]

4x[(x - 3)(3x + 2)]Tuesday, March 3, 2009

Example 4: Factor.

a. x2 + 5x + 6 b. x2 - 7x + 6


Example 4: Factor.

a. x2 + 5x + 6 b. x2 - 7x + 6(1)(6) = 6


Example 4: Factor.

a. x2 + 5x + 6 b. x2 - 7x + 6(1)(6) = 6

2 + 3 = 5


Example 4: Factor.

a. x2 + 5x + 6

(x + 2)(x + 3)

b. x2 - 7x + 6(1)(6) = 6

2 + 3 = 5


Example 4: Factor.

a. x2 + 5x + 6

(x + 2)(x + 3)

b. x2 - 7x + 6(1)(6) = 6

2 + 3 = 5

(1)(6) = 6


Example 4: Factor.

a. x2 + 5x + 6

(x + 2)(x + 3)

b. x2 - 7x + 6(1)(6) = 6

2 + 3 = 5

(1)(6) = 6

-1 - 6 = -7


Example 4: Factor.

a. x2 + 5x + 6

(x + 2)(x + 3)

b. x2 - 7x + 6

(x - 1)(x - 6)

(1)(6) = 6

2 + 3 = 5

(1)(6) = 6

-1 - 6 = -7


Example 5: Factor.a. 4x2 - 16x + 16 b. 25b4 - 81r8

c. 2x2 + x - 6 d. 4x2 - 19x + 12



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

2x2 + 4x - 3x - 6



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

(x + 2)(2x - 3)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

(x + 2)(2x - 3)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)

4x2 - 16x - 3x + 12



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

(x + 2)(2x - 3)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)

4x2 - 16x - 3x + 12(4x2 - 16x) + (- 3x + 12)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

(x + 2)(2x - 3)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)

4x2 - 16x - 3x + 12(4x2 - 16x) + (- 3x + 12)

4x(x - 4) - 3(x - 4)



c. 2x2 + x - 6 d. 4x2 - 19x + 12

4(x - 2)2 (5b2 - 9r4)(5b2 + 9r4)

(x + 2)(2x - 3)

2x2 + 4x - 3x - 6(2x2 + 4x) + (-3x - 6)2x(x + 2) - 3(x + 2)

4x2 - 16x - 3x + 12(4x2 - 16x) + (- 3x + 12)

4x(x - 4) - 3(x - 4)(x - 4)(4x - 3)


e. 2x2 - 4x - 16 f. -6x2 - x + 2


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]

-1(6x2 + x - 2)


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]

-1(6x2 + x - 2)-1(6x2 + 4x - 3x - 2)


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]

-1(6x2 + x - 2)-1(6x2 + 4x - 3x - 2)

-[(6x2 + 4x) + (- 3x - 2)]


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]

-1(6x2 + x - 2)-1(6x2 + 4x - 3x - 2)

-[(6x2 + 4x) + (- 3x - 2)]-[2x(3x + 2) -1(3x + 2)]


e. 2x2 - 4x - 16 f. -6x2 - x + 22(x2 - 2x - 8)

2(x2 + 2x - 4x - 8)2[(x2 + 2x) + (-4x - 8)]2[x(x + 2) - 4(x + 2)]

2[(x + 2)(x - 4)]

-1(6x2 + x - 2)-1(6x2 + 4x - 3x - 2)

-[(6x2 + 4x) + (- 3x - 2)]-[2x(3x + 2) -1(3x + 2)]

-(3x + 2)(2x - 1)


Homework


Homework

You’ll know it when Mr. Lamb passes it out.


AA Section 11-3 Day 2

Education