Question

Which polynomial represents the product of
2x-3 and x^2+6x-2
?
A
2x3+6x−62x^{3}+6x-62x
3
+6x−6
B
2x3−18x−22x^{3}-18x-22x
3
−18x−2
C
2x3+9x2−22x+62x^{3}+9x^{2}-22x+62x
3
+9x
2
−22x+6
D
2x3+15x2−14x+62x^{3}+15x^{2}-14x+62x
3
+15x
2
−14x+6

Answer 1

To find the product of the polynomials \(2x - 3\) and \(x^2 + 6x - 2\), we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions.

The polynomial multiplication can be computed as follows:

\[
(2x - 3)(x^2 + 6x - 2)
\]

We can expand this:

1. Multiply \(2x\) by each term in the second polynomial:
- \(2x \cdot x^2 = 2x^3\)
- \(2x \cdot 6x = 12x^2\)
- \(2x \cdot (-2) = -4x\)

2. Multiply \(-3\) by each term in the second polynomial:
- \(-3 \cdot x^2 = -3x^2\)
- \(-3 \cdot 6x = -18x\)
- \(-3 \cdot (-2) = 6\)

Now we can combine all the results:

\[
2x^3 + 12x^2 - 4x - 3x^2 - 18x + 6
\]

Next, we will combine like terms:

- For \(x^2\): \(12x^2 - 3x^2 = 9x^2\)
- For \(x\): \(-4x - 18x = -22x\)

Putting it all together, we have:

\[
2x^3 + 9x^2 - 22x + 6
\]

Thus, the resulting polynomial is:

\[
2x^3 + 9x^2 - 22x + 6
\]

Now we can match this result with one of the provided options:

C: \(2x^3 + 9x^2 - 22x + 6\)

Therefore, the correct answer is **C**.