Answer :
Sure, let's simplify the expression [tex]\((4x - 3)(3x^2 - 4x - 3)\)[/tex] step-by-step.
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, distribute [tex]\(4x\)[/tex] to each term in [tex]\(3x^2 - 4x - 3\)[/tex]:
[tex]\[
4x \cdot 3x^2 = 12x^3
\][/tex]
[tex]\[
4x \cdot (-4x) = -16x^2
\][/tex]
[tex]\[
4x \cdot (-3) = -12x
\][/tex]
- Now, distribute [tex]\(-3\)[/tex] to each term in [tex]\(3x^2 - 4x - 3\)[/tex]:
[tex]\[
-3 \cdot 3x^2 = -9x^2
\][/tex]
[tex]\[
-3 \cdot (-4x) = 12x
\][/tex]
[tex]\[
-3 \cdot (-3) = 9
\][/tex]
2. Combine all the distributed terms:
[tex]\[
12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-16x^2 - 9x^2 = -25x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-12x + 12x = 0x
\][/tex]
4. Write the simplified expression:
[tex]\[
12x^3 - 25x^2 + 9
\][/tex]
So, the correct simplification of [tex]\((4x - 3)(3x^2 - 4x - 3)\)[/tex] is [tex]\(12x^3 - 25x^2 + 9\)[/tex], which corresponds to the answer choice:
[tex]$12x^3 - 25x^2 + 9$[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, distribute [tex]\(4x\)[/tex] to each term in [tex]\(3x^2 - 4x - 3\)[/tex]:
[tex]\[
4x \cdot 3x^2 = 12x^3
\][/tex]
[tex]\[
4x \cdot (-4x) = -16x^2
\][/tex]
[tex]\[
4x \cdot (-3) = -12x
\][/tex]
- Now, distribute [tex]\(-3\)[/tex] to each term in [tex]\(3x^2 - 4x - 3\)[/tex]:
[tex]\[
-3 \cdot 3x^2 = -9x^2
\][/tex]
[tex]\[
-3 \cdot (-4x) = 12x
\][/tex]
[tex]\[
-3 \cdot (-3) = 9
\][/tex]
2. Combine all the distributed terms:
[tex]\[
12x^3 - 16x^2 - 12x - 9x^2 + 12x + 9
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-16x^2 - 9x^2 = -25x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-12x + 12x = 0x
\][/tex]
4. Write the simplified expression:
[tex]\[
12x^3 - 25x^2 + 9
\][/tex]
So, the correct simplification of [tex]\((4x - 3)(3x^2 - 4x - 3)\)[/tex] is [tex]\(12x^3 - 25x^2 + 9\)[/tex], which corresponds to the answer choice:
[tex]$12x^3 - 25x^2 + 9$[/tex].
