Math  /  Algebra

QuestionFactor the trinomial completely. x2x56x^{2}-x-56
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. x2x56=\mathrm{x}^{2}-\mathrm{x}-56= \square (Type your answer in factored form.) B. The polynomial is prime.

Studdy Solution

STEP 1

What is this asking? We need to rewrite this *trinomial* x2x56x^2 - x - 56 as a product of simpler expressions, if possible.
If it's not possible, we'll declare it *prime*. Watch out! Don't rush!
Carefully consider the signs of the terms.
A common mistake is messing up the plus and minus signs when factoring.

STEP 2

1. Find the magic numbers
2. Verify the magic numbers
3. Factor the trinomial

STEP 3

Alright, let's **find two numbers** that **multiply** to 56-56 (the **constant term**) and **add** up to 1-1 (the **coefficient** of the xx term).
This is like a fun little puzzle!

STEP 4

Let's think about the factors of 56-56.
We could have 11 and 56-56, 1-1 and 5656, 22 and 28-28, 2-2 and 2828, and so on.
Keep going!
We also have 77 and 8-8, and 7-7 and 88.

STEP 5

Which of these pairs add up to 1-1?
It's 77 and 8-8! 7+(8)=17 + (-8) = -1.
These are our **magic numbers**!

STEP 6

Let's double-check our magic numbers just to be sure.
We found 77 and 8-8.
Do they multiply to 56-56?
Yes! 7(8)=567 \cdot (-8) = -56.
Do they add up to 1-1?
Yes! 7+(8)=17 + (-8) = -1.
Perfect!

STEP 7

Now, we can rewrite our original expression using these magic numbers.
We have x2x56x^2 - x - 56.
Using our magic numbers, 77 and 8-8, we can write this as (x+7)(x8)(x + 7)(x - 8).

STEP 8

Let's **expand** this just to be absolutely certain: (x+7)(x8)=xx+x(8)+7x+7(8)=x28x+7x56=x2x56(x+7)(x-8) = x \cdot x + x \cdot (-8) + 7 \cdot x + 7 \cdot (-8) = x^2 - 8x + 7x - 56 = x^2 - x - 56.
Boom! We're back where we started.

STEP 9

So, the factored form of x2x56x^2 - x - 56 is (x+7)(x8)(x+7)(x-8).
We choose **A** and fill in the box with (x+7)(x8)(x+7)(x-8).

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