Question

Select all of the factors for the given polynomial:

−18k2+15k+168


(3k - 42)


(6k - 42)


(3k + 48)


(2k - 7)


(3k - 8)


-3


-6


(6k + 48)


(3k + 8)

Answer 1

To factor the polynomial \(-18k^2 + 15k + 168\), let's first look for the greatest common factor (GCF). The GCF of the coefficients \(-18\), \(15\), and \(168\) is \(3\). We can factor \(3\) out of the polynomial:

\[
-18k^2 + 15k + 168 = 3(-6k^2 + 5k + 56)
\]

Next, we need to factor the quadratic expression \(-6k^2 + 5k + 56\). To do this, we can use the method of factoring by grouping or by applying the quadratic formula. However, let's first rewrite the quadratic in a more manageable form:

\[
-6k^2 + 5k + 56 = -1(6k^2 - 5k - 56)
\]

Now we'll focus on factoring \(6k^2 - 5k - 56\). This requires finding two numbers that multiply to \(6 \times -56 = -336\) and add up to \(-5\). The correct pair of numbers is \(16\) and \(-21\):

Now we can express the middle term:

\[
6k^2 - 21k + 16k - 56
\]

Group the terms:

\[
(6k^2 - 21k) + (16k - 56)
\]

Factor out common factors:

\[
3k(2k - 7) + 8(2k - 7) = (2k - 7)(3k + 8)
\]

Thus, the original polynomial factors to:

\[
-18k^2 + 15k + 168 = 3 \cdot -1 \cdot (2k - 7)(3k + 8) = -3(2k - 7)(3k + 8)
\]

Now that we have the factors as \( -3\), \((2k - 7)\), and \((3k + 8)\), we can compare these with the options provided.

The factors of the given polynomial \(-18k^2 + 15k + 168\) are:

- \((2k - 7)\)
- \( -3\)
- \((3k + 8)\)

From your list, the factors that match are:

- (2k - 7)
- -3
- (3k + 8)

Thus, the selected factors for the polynomial are:

- \((2k - 7)\)
- \(-3\)
- \((3k + 8)\)