Question

Kathy wants to factor x2 - 12x + 36 by grouping; however, Tom says it is a special product and can factor a different way. Using complete sentences explain and demonstrate how both methods will result in the same factors

Answer 1

A special product that has three terms, such as this one, usually is the square of a binomial.
To check to see if a trinomial is the square of a binomial, do this:
1. Is the first term the square of something? If so, what?
2. Is the last term the square of something? If so, what?
3. Multiply the quantities you got in 1 and 2 above and then multiply by 2. Do you get the middle term? If so, it's the square of a binomial.

Answer 2

Now let's do it together.

Answer 3

okay :)

Answer 4

We're looking at the possibility it is a special product first. We'll do the factoring by grouping later.

Answer 5

Let's go through those questions.

Answer 6

It's helpful to keep our problem in mind:
x^2 - 12x + 36

1. Is the first term the square of something? If so, what?
x^2 is the square of x

Answer 7

The first terms " the square of x by x "
then the second question no

Answer 8

2. Is the last term the square of something? If so, what?
36 is the square of 6 and -6

Answer 9

See above. x^2 is the square of x, so the answer is yes, and we get x from question 1.

Answer 10

36 is the square of 6 and -6, so the answer to question 2 is yes, and the numbers are 6 or -6.

Answer 11

3. Now take the answers to questions 1 and 2, multiply them together and multiply by 2.
If we use x and 6, we get x * 6 * 2 = 12x
If we use x and - 6, we get x * (-6) * 2 = -12x

Answer 12

Now we compare 12x and -12x to the middle term of the trinomial we need to factor, and we see that -12x is indeed the middle term of the original problem, so the trinomial really is a special product, and it is:
x^2 - 12x + 36 = (x - 6)^2

Answer 13

Tom is correct. This trinomial is a special product. It is the square of a binomial.

Answer 14

Now let's go back to the original trinomial to factor it by grouping.

Answer 15

To factor a trinomial of the form ax^2 + bx + c by grouping, follow these steps:
1. Multiply ac together.
2. Find two factors of ac that add to b. Let's call them p and q.
3. Break up bx into px + qx, so you now have ax^2 + px + qx + c.
4. Factor the 4-term polynomial by grouping. That means, factor a common factor out of the first two terms. Factor a common factor out of the last two terms.
5. Factor out the common factor.

Answer 16

Now let's factor our trinomial by groupioing following the steps above.
1. ac = 1 * 36 = 36
2. (-6) * (-6) = 36, and -6 + (-6) = -12, so our p and q are -6 and -6.
3. x^2 - 6x - 6x + 36
4. x(x - 6) - 6(x - 6)
5. (x - 6)(x - 6) = (x - 6)^2