In order to factor x ^ 2 – 2x – 8 we must break up the “bx” term into two terms whose coefficients…
add to-2 and multiply to 8.
add to 8 and multiply to 2.
add to 1 and multiply to -8.
add to 1 and multiply to 2.
The correct answer and explanation is:
The correct answer is add to -2 and multiply to -8.
Explanation:
When factoring a quadratic expression of the form ax^2 + bx + c, the goal is to find two numbers that multiply to ac and add to b. In this case, the quadratic expression is x^2 – 2x – 8. Here, a = 1, b = -2, and c = -8.
To factor this quadratic, we need to find two numbers that:
- Multiply to a * c = 1 * (-8) = -8.
- Add up to b = -2.
Now, let’s find two numbers that meet these conditions:
- The factors of -8 are:
- (1, -8)
- (-1, 8)
- (2, -4)
- (-2, 4)
Out of these pairs, -4 and 2 add up to -2. Therefore, these are the numbers that we need to break up the bx term.
Step-by-step process:
- Start with the original quadratic: x^2 – 2x – 8.
- Break up the bx term -2x into -4x + 2x: x2−4x+2x−8x^2 – 4x + 2x – 8
- Factor by grouping:
- Group the first two terms: x^2 – 4x.
- Group the last two terms: 2x – 8.
- Factor out the greatest common factor (GCF) from each group:
- From x^2 – 4x, factor out x: x(x – 4).
- From 2x – 8, factor out 2: 2(x – 4).
- Now, factor out the common binomial factor (x – 4): (x−4)(x+2)(x – 4)(x + 2)
Thus, the factored form of x^2 – 2x – 8 is (x – 4)(x + 2).
This demonstrates the factoring process and shows that the correct numbers to split the bx term were -4 and 2, which add up to -2 and multiply to -8.