Factor each polynomial- x^2 -6x + 9 =?

Question

I'm more interested in the steps in getting the answer than the answer itself. Thanks =)

btw thats
x^2 -6x + 9 =

Answer 1

x^2 -6x + 9 =
(x-3) (x-3) =
(x-3)^2

Now let me explain why I did those things... you see, you need to use the method F.O.I.L (First, Outside, Inside, Last).

We'll start with the F, first. Look at the first term: x^2. What two things can you multiply together so that you get x^2? x and x of course! So that's the First term in each) of your two factors. So far we have

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

Next, I like to skip right ahead to the Last term, because that narrows down your options so that you're not doing a huge stupid trial and error thing.... Look at 9. What two things can be multiplied together to get 9? There is 1 and 9, and 3 and 3. Since 9 is positive, we know that BOTH of the factors in each the 1 and 9, and the 3 and 3 have to BOTH be either positive or negative. (Remember, because negative X negative = positive, and of course positive X positive = positive, but negative X positive = negative, and positive X negative = negative). So we know that our options are one of the following:

(x+3)(x+3)

OR:

(x-3)(x-3)

OR:

(x+1)(x+9)

OR:

(x-1)(x-9)

Now, we look at the Outside and Inside. That means that the outside terms (multiplied by each other), added the inside terms (multiplied by each other), must equal -6x (because that is the MIDDLE TERM).

For example, for:

(x+3)(x+3)

You do the OUTSIDE numbers (the first x and the last 3)

(x+3)(x+3)
^ ........... ^

Which would give you 3x (positive).

Next, you do the INSIDE numbers (the first 3 and the second x):

(x+3)(x+3)
.....^. ^

Which again, gives you 3x (positive).

When we add the answers together, (3x + 3x), you get +6x. This is close, but NOT the right answer because we are looking for NEGATIVE 6x. But, since the coefficient is correct, we know that we are using the right set of factors (3 and 3, instead of the 1 and 9).

What is our only other option for an equation involving 3 and 3 as factors?

It's: (x-3)(x-3)

Let's factor it out to check!


First:

(x-3)(x-3)
^ ..... ^

x^2

Outside:

(x-3)(x-3)
^ ....... ^

-3x

Inside:

(x-3)(x-3)
.....^..^

-3x

(Add them together [-3x + -3x = -6x])

Last:

(x-3)(x-3)
.....^......^

-3 X -3 = 9


When we put the equation together: First, Outside+Inside, Last, we get the original equation: x^2 -6x + 9

Perfect! So we know that (x-3)(x-3) is the right choice!

I hope this helps!!!!!!

Answer 2

(x-3)*(x-3)

I do this mostly by recognizing the form. A polynomial of this configuration is going to be the product of (x+/-n) * (x+/-n).
You see this by noting that the first term in the polynomial is x^2. Thats the product of x * x. The last term is +9. That's probably the product of 3*3 or (-3) *(-3) or 1 * 9. The middle term has to be the sum of ________. The sum of (-3) and (-3) is (-6).
Then you multiply (x-3) by (x-3) you get x^2 + (-3)x + (-3)x + (-3)*(-3). Simplifying you get x^2 -6x +9.

Answer 3

The process I use to polynomials is:

You already know you will have -
( x + _ )( x + _ )
To figure out the terms that belong in the blank, look at 'b' which is -6 the coefficient to x, and 'c' which is 9. (ax^2 + bx + c)
Know I think in my head, what factors multiply to make 9, and add together to make -6.

If you are smart enough you will visualize -3 and -3, factors of 9, and add together -6

So the answer is:
( x - 3 )^2

Answer 4

product = 9
sum = 6
factor = -3 , -3
:.x^2 -6x + 9 = (x-3)(x-3)

Answer 5

x^2 -6x + 9
= x^2 -3x - 3x +9
= x(x-3) -3(x-3)
= (x-3)(x-3)
= (x-3)^2

Answer 6

x^2-6x+9=0
(x-3)(x-3)=0 (x-3)^2=0
x=3

Answer 7

= (x - 3).(x - 3) = (x - 3)²