Question

What is the product of the two polynomials shown? (3x^2−4x)(9x^4+12x^3+16x^2)

Answer 1

To find the product of the two polynomials \( (3x^2 - 4x)(9x^4 + 12x^3 + 16x^2) \), we will distributively multiply each term in the first polynomial by each term in the second polynomial.

Starting with the first polynomial \( 3x^2 - 4x \):

1. **Multiply \( 3x^2 \) by each term in \( 9x^4 + 12x^3 + 16x^2 \):**
- \( 3x^2 \cdot 9x^4 = 27x^6 \)
- \( 3x^2 \cdot 12x^3 = 36x^5 \)
- \( 3x^2 \cdot 16x^2 = 48x^4 \)

2. **Multiply \( -4x \) by each term in \( 9x^4 + 12x^3 + 16x^2 \):**
- \( -4x \cdot 9x^4 = -36x^5 \)
- \( -4x \cdot 12x^3 = -48x^4 \)
- \( -4x \cdot 16x^2 = -64x^3 \)

Now, combine all these terms:

\[
27x^6 + 36x^5 + 48x^4 - 36x^5 - 48x^4 - 64x^3
\]

Next, we will combine like terms:

- The \( x^5 \) terms: \( 36x^5 - 36x^5 = 0 \)
- The \( x^4 \) terms: \( 48x^4 - 48x^4 = 0 \)
- The \( x^3 \) term: \( -64x^3 \) remains as is.

The remaining terms are:

\[
27x^6 - 64x^3
\]

Thus, the product of the two polynomials is:

\[
\boxed{27x^6 - 64x^3}
\]