Please Help Me Factor?

Question

a)

x^2 - 12x + 11

b)

x^2 + 6x + 5

c)

x^2 + 2x +1

d)

x^2-16x + 64

e)

x^2+x-12

f)

x^2 + 7x – 30

Answer 1

Factoring is a piece of cake!

Factoring is the opposite or reverse of multiplying. Having skill in multiplying is having skill in factoring. This specifically is factoring a trinomial. I'll give you an example and I will also help you with one of the ones you listed.

Here is the example:

2x² + 9x − 5

-- it will be factored as a product of binomials:

(?   ?)(?   ?)

Now,  how will 2x² be produced?  There is only one way:  2x· x :

(2x   ?)(x   ?)

And how will 5 be produced?  Again, there is only one way:  1·  5.  But does the 5 go with  2x  or with  x ?

(2x   5)(x   1)    or     (2x   1)(x   5) ?

Notice:  We have not yet placed any signs!

How shall we decide between these two possibilities?  It is the combination that will correctly give the middle term, 9x :

2x² + 9x − 5.

Consider the first possibility:

(2x   5)(x   1)

Is it possible to produce  9x  by combining the outers and the inners:  2x· 1 with 5x ?

No, it is not.  Therefore, we must eliminate that possibility and consider the other:

(2x   1)(x   5)

Can we produce  9x  by combining  10x  with x ?

Yes -- if we choose +5 and −1:

(2x − 1)(x + 5)

(2x − 1)(x + 5) = 2x² + 9x − 5

Skill in factoring depends on skill in multiplying -- particularly in picking out the middle term!

________________________________

Now about your first problem (a).

We have here x^2 -12x+11

1) (? ?)(? ?)  <--- start by making your parenthasies.

Now look at x^2. What is the only way it can be produced?

----> by multiplying x and x, so that means you have:

(x ?)(x ?)

2) Next you want to figure out how the 11 can be produced.

Here are your possibilities:

     11 times 1

Thats it. Thats the only possibility.

Now you have to make sure that the 11 and 1 can somehow make -12, the second coefficieant. It can if you make the signs on 11 and 1 both negative (-).

[HINT: if your first sign is a (-) and your second is a (+) then you will have two negatives (?-a)(?-b). The first sign will become the sign of your bigger factor. The second sign tells you wether the signs will be the same or not. So if the first sigh is (+) then your bigger number will be positive. If your second sign is a (-) then you will have different signs( a + for the bigger and a - for the smaller). Get it?]

So here is what we have:

(x-1)(x-11)

So now we know that its:

(x-1)(x-11) = x^2-12x+11

Check your work by using F.O.I.L. which stands for First Outside Inside and Last.

First, multiply x times x. You should get x^2. Okay check on that.

Then for Outside multiply the outsides x times -11 and you should get -11x. Good. [so far its x^2-11x]

Then do Inside by multiplying -1 times x. You should get -1x. [now its x^2-11x-1x]

The last thing is to multiply the last ones which is -1 times -11 = 11 because a negative times a negative equal a positive.

[so now it is x^2-11x-1x+11]

Almost done checking...

Now just combine like terms, which would be the -11x and -1x. Add them and you get -12x, YOUR MIDDLE TERM!

GREAT!! Now your checking comes down to x^2-12x+11. !!! ;)

:) :) :)

And thats it, your done. Now wasnt that simple?

Now try the rest of them on your own.

Answer 2

a.) x^2 - 12x + 11

=(x-11)(x-1)

b.) x^2 + 6x + 5

=(x+5)(x+1)

c.)x^2 + 2x +1

=(x+1)(x+1)

d.)x^2-16x + 64

=(x-8)(x-8)

e.)x^2+x-12

=(x+4)(x-3)

f.)x^2 + 7x – 30

=(x-3)(x+10)

hope this helps

Answer 3

a)

=x^2 - 12x + 11

=x^2 - 11x - x + 11

= x( x-11) - 1(x - 11)

= (x-1)(x-11)

b)

x^2 + 6x +5

=x^2 + 5x + x + 5

=x( x+5) + 1( x+5)

=( x+5) (x+1)

c)

=x^2 + 2x + 1

=x^2 + x + x + 1

=x( x+1) + 1 (x + 1)

=(x+1)(x+1)

=( x+1)^2

d)

=x^2 -16x + 64

=x^2 - 8x -8x + 64

=(x-8)^2

e)

=x^2 + x -12

=x^2 + 4x - 3x - 12

=(x-4)(x+3)

f)

=x^2 + 7x - 30

=x^2 + 10x -3x - 30

=(x+10)(x-3)

                        

Hope I Helped    (^_^)

Answer 4

a.) (x-11)(x-1)

b.)(x+3)(x+2)

c.) (x+1)(x+1)

d.) (x-8)(x-8)

e.) (x+4)(x-3)

f.) (x+10)(x-3)

Answer 5

a. (x-11) (x-1)

b. (x+5) (x+1)

c. (x+1) (x+1)

d. (x-8)   (x-8)

e. (x+4) (x-3)

f. (x+10) (x-3)

Answer 6

a)  (x-11)(x-1)

b)  (x+5)(x+1)

c)  (x+1)^2

d) (x-8)(x-8)

e)  (x+4)(x-3)

f)  (x+10)(x-3)

Answer 7

a)

x^2 - 12x + 11

You know the result will be in the form

(x ____) (x _____)

First, look at the sign in front of the constant (the 11 in this case).  If it's a +, you know both the signs in the final will be the same (either both + or both -).

Now, look at the sign in front of the regular x (the 12x in this case).  That sign is -, so that means both the signs in the final will be -.

So, now we know the answer will look like this:

(x - ??)(x - ??)

Now... Go back to the constant (11).  What are the factors of that number?

11 = 11 * 1

Do those factors add to get the number in front of the x? (In this case, 12?)

Yes, they do.

So, our answer is

(x - 11)(x - 1)

To double check... Multiply it back out.

(x - 11)(x - 1)

= x(x - 1) - 11(x - 1)

= x^2 - x - 11x + 11

= x^2 - 12x + 11

That's what we started with, so that's the answer.

b)

x^2 + 6x + 5

The second sign is a +, so we know both signs must be + or both are -.  The first sign (in front of the 6x) is +, so both signs are +.

(x + ??)(x + ??)

What are the factors of 5?  (5 * 1)

Do they add to 6?  (yes)

(x + 5)(x + 1) <---- solution

c)

x^2 + 2x +1

= (x + 1)(x + 1)

d)

x^2-16x + 64

(x - ??)(x - ??)

Factors of 64:

1 * 64

2 * 32

4 * 16

8 * 8

Which set adds to 16?  (8 * 8)

So, solution is

(x - 8)(x - 8)

= (x - 8)^2

e)

x^2+x-12

Now this one is different.  The sign in front of the constant (12) is -.  That means one sign must be - and one sign must be +.

(x - ??)(x + ??)

Factors of 12:

1 * 12

2 * 6

3 * 4

Now... Since we have one of each sign (+ and -)... Which set of factors SUBTRACTS to get POSITIVE 1?  (Positive because that's the sign in front of the regular x).

4 - 3 = 1

Notice that the 4 is positive and the 3 is negative.  

So the answer is

(x - 3)(x + 4)

f)

x^2 + 7x – 30

(x - ??)(x + ??)

Factors of 30?

1 * 30

2 * 15

3 * 10

Which set SUBTRACTS to get POSITIVE 7?

10 - 3

So...(x + 10)(x - 3)

Answer 8

a)

x^2 - 12x + 11

= x^2 - x - 11x + 11

= (x^2 - x) - (11x - 11)

= x(x - 1) - 11(x - 1)

= (x - 1)(x - 11)

b)

x^2 + 6x + 5

= x^2 + 5x + x + 5

= (x^2 + 5x) + (x + 5)

= x(x + 5) + 1(x + 5)

= (x + 5)(x + 1)

c)

x^2 + 2x + 1

= x^2 + x + x + 1

= (x^2 + x) + (x + 1)

= x(x + 1) + 1(x + 1)

= (x + 1)(x + 1)

= (x + 1)^2

d)

x^2 - 16x + 64

= x^2 - 8x - 8x + 64

= (x^2 - 8x) - (8x - 64)

= x(x - 8) - 8(x - 8)

= (x - 8)(x - 8)

= (x - 8)^2

e)

x^2 + x - 12

= x^2 + 4x - 3x - 12

= (x^2 + 4x) - (3x + 12)

= x(x + 4) - 3(x + 4)

= (x + 4)(x - 3)

f)

x^2 + 7x - 30

= x^2 + 10x - 3x - 30

= (x^2 + 10x) - (3x + 30)

= x(x + 10) - 3(x + 10)

= (x + 10)(x - 3)

Answer 9

x^2 - 12x + 11

= (x - 11)(x - 1)

x^2 + 6x + 5

= (x + 5)(x + 1)

x^2 + 2x + 1

= (x + 1)(x + 1)

= (x + 1)^2

x^2 - 16x + 64

= (x - 8)(x - 8)

= (x - 8)^2

x^2 + x - 12

= (x + 4)(x - 3)

x^2 + 7x - 30

= (x - 3)(x + 10)

Answer 10

a)

x^2 - 12x + 11

Factors of +11 that when added will have -12 as the sum: -11 and -1

Therefore, x^2 - 12x + 11 = (x - 11)(x - 1)

b)

x^2 + 6x + 5

Factors of +5 that when added will have +6 as the sum: +5 and +1

Therefore, x^2 + 6x + 5 = (x + 5)(x + 1)

c)

x^2 + 2x +1

Factors of +1 that when added will have +2 as the sum: +1 and +1

Therefore, x^2 + 2x + 1 = (x + 1)(x + 1) = (x + 1)^2

d)

x^2-16x + 64

Factors of +64 that when added will have -16 as the sum: -8 and -8

Therefore, x^2 - 16x + 64 = (x - 8)(x - 8) = (x - 8)^2

e)

x^2+x-12

Factors of -12 that when added will have +1 as the sum: +4 and -3

Therefore, x^2 + x -12 = (x + 4)(x - 3)

f)

x^2 + 7x – 30

Factors of -30 that when added will have +7 as the sum: +10 and -3

Therefore, x^2 + 7x - 30 = (x + 10)(x - 3)