Question

Part 1: Joe and Hope were both asked to factor the following polynomial completely. Is one of them correct? Both of them? Neither of them? Explain what each of them did was correct and/or incorrect.

Part 2:

Find all the values of k so the the quadratic expression factors into two binomials. Explain the process used to find the values.

3x^2 + kx - 8​

Answer 1

Hope this attachment helps you.

Answer 2

Part 1

If we simplify, both Joe and Hope factored the polynomial correctly but Joe didn't complete it fully.

The first binomial can be further factored:

• 8x + 12 = 4(2x + 3)

Part 2

The quadratic expression needs to have two roots in order to be factored as two binomials.

The discriminant must be positive or zero:

• D = b² - 4ac ≥ 0

We have a = 3, b = k, c = -8

So we get following inequality:

• k² - 4*3*(-8) ≥ 0
• k² + 96 ≥ 0

Since k² is positive for any value of k, the solution is any value of k:

• k ∈ R