Algebra 1 > Notes > Yorkcounty Final > Yorkcounty > Factoring - Continued

Practice • Factor the expression completely x2 – 2x –15

• answer on next slide

Answer • x2 –2x –15 • First we need to find the factors of –15 • -1, 15 1, -15 • -3, 5 3, -5 • Which of these pairs add to –2? 3 and –5 • So put down your parentheses with the x’s in them. • (x )(x ) now put in you factors • (x + 3)(x - 5) this is our factored expression

Solving Quadratics • x2 + 7x + 10 = 0 • We want to solve this quadratic equation. The first thing you have to do is factor the left side of the equation. • You need two numbers that multiply to positive 10 that add to positive 7. • Factors of 10. 1,10 -1,-10 2,5 -2,-5 • 2,5 is your pair. 2 times 5 equals 10 2 plus 5 equals 7 • (x + 2)(x + 5) = 0 Now that we have factored the left side we need to finish solving the equation. What you have to do is split each of the binomials apart and set them both equal to zero. • (x + 2) = 0 and (x + 5) = 0 Next you have to solve for x • next slide…..

(solve for x means get x by itself on one side)

Solving • (x + 2) = 0 • subtract 2 from both sides • x+2–2=0–2 • x = -2

and

(x + 5) = 0 subtract 5 from both sides x+5–5=0-5 x = -5

• Our two solutions are –2 and –5. • Next we want to check our solutions. Take each of your solutions and check them separately by substituting them for x in the ORIGINAL equation. If your solutions are correct you will make a true statement. • next slide…..

Solving – Checking Solutions • x2 + 7x + 10 = 0 • we found two solutions to be –2 and –5 • • • • • • •

Substitute –2 for x. Substitute –5 for x (-2)2 + 7(-2) + 10 = 0 (-5)2 + 7(-5) + 10 = 0 4 – 14 + 10 = 0 25 - 35 + 10 = 0 14 – 14 = 0 35 – 35 = 0 0 = 0 (true statement) 0 = 0 (true statement) -2 works -5 works Our two solutions are –2 and –5

Try this • Solve:

•

x2 + 7x – 18 = 0

• Next slide for solution

x2 + 7x – 18 = 0 • Find the factors of –18 -1, 18 1, -18 -2, 9 2, -9

-3, 6

3, -6

Which of these pairs multiply to –18 and add to +7 ? -2 times 9 equals –18 and –2 plus 9 equals 7 (x – 2)(x + 9) break apart the binomials x–2=0 x+9=0 x–2+2=0+2 x+9–9=0–9 x=2 x = -9 Check your solutions

next slide…

-2 and 9

x2 + 7x – 18 = 0 • • • • •

Check your solutions (2)2 + 7(2) – 18 = 0 4 + 14 – 18 = 0 18 – 18 = 0 0 = 0 2 works!

2 and –9 (-9)2 + 7(-9) – 18 = 0 81 – 56 – 18 = 0 81 – 81 = 0 0 = 0 -9 works

• Our two solutions are 2 and –9

“Factor the trinomial: 2x2 -13x + 15” This problem the coefficient of x2 is greater than one. We now have to use a new method to factor. The first term is 2x2. The only way to multiply with x’s and get that is 2x times 1x, or -2x times -1x. The last term is +15. The only way to multiply and get 15 is to use 3 and 5 or -3 and -5.

Let’s try 2x and 1x for the 2x2. And let’s try 3 and 5 for the +15. 2x

1x

3

5

=

2x2

= 15

Now we need to check and see if the middle term equals -13x.

“Factor the trinomial: 2x2 -13x + 15” To check the middle term we need to cross multiply and see if our two answers add up to -13x.

3x

2x

1x

=

2x2

3

5

= 15

10x

Unfortunately they do not, 3x and 10x make positive 13x and not negative 13 x. So, we can try using another one of our possibilities, like -3 and -5 for +15.

-3x

2x

1x

=

2x2

-3

-5

= 15

-10x

This works, -3x and -10x make -13 x. We can now check our answer just to make sure it is correct--see the next slide.

2x

1x

=

2x2

-3

-5

= 15

Since 2x and -3 are vertical together we write (2x -3) Since 1x and -5 are vertical together we write (x - 5) Answer:

(2x - 3) (x - 5)

Check:

2x (x -5) -3 (x - 5) 2x2 - 10x - 3x + 15 2x2

- 13x

+ 15

Since this matches the given trinomial exactly, our answer of (2x-3)(x-5) is correct.

Example 1: Factor 3y2 + 7y - 20. A) (3y-5)(y - 4) B) (3y-2)(y+10) C) (3y-4)(y+5) D) (3y-5)(y+4)

Question: What should we use for factors of 3y2. What are the possibilities of factors for -20? Write them down and then advance to the next slide to see the rest of the solution.

Hopefully you said to use 3y and y for 3y2. The possibilities for -20 are: -10,2

5, -4

10,-2

-5,4

So, that gives us four possibilities to try out: You perform the cross-multiplication and then advance to check. 3y

y = 3y2

3y

y = 3y2

-10

2 = -20

10

-2 = -20

3y

y = 3y2

3y

y = 3y2

3y

y = 3y2

3y

y = 3y2

-10

2 = -20

10

-2 = -20

-10y

5y

6y

3y

y = 3y2

5

-12y -4 = -20

10y

-5y

-6y

3y

y = 3y2

-5

12y 4 = -20

Which possibility works?????????????????????????????????????

-5y

3y

y = 3y2

-5

4 = -20

Yes, this is the only one in which the two cross terms add up to +7y, the middle term of the polynomial.

12y

Thus, the answer is the vertical binomial (3y - 5) times the other vertical binomial (y + 4) The best choice is D. Note: If none of the possibilities work out, try switching the bottom terms, for example if the one at the top of this slide didn’t work, try switching around the -5 and 4 and working them again.

Try These:

1) Factor x2 - 7x + 12

2) Factor 3x2 -14x + 8

3) Factor 2x2 -x -10

Try This Solutions: 1) x2 -7x + 12

Use x and x for x2 Possibilities for +12 are: 6,2

-6, -2

3,4

-3,-4

This is the one that works x

x

= x2

So the answer is (x-3)(x-4)

-3

-4

-3x add to -7x

-4x

= +12

Try This Solutions: 2) 3x2 -14x + 8

Use 3x and x for3 x2 Possibilities for +8 are: 8,1

-8, -1

2,4

-2,-4

This is the one that works 3x

x

= x2

So the answer is (3x-2)(x-4)

-2 -2x

-4 -12x

add to -14x

= +12

Try This Solutions: 3) 2x2 -x -10

Use 2x and x for 2x2 Possibilities for -10 are: -5,2

5, -2

-10,1

10,-1

This is the one that works 2x

x

= x2

So the answer is (2x-5)(x+2)

-5

2

-5x add to -1x

4x

= +12