Answer :
To find the product of the polynomials [tex]\(2x^4(4x^2 + 3x + 1)\)[/tex] and [tex]\(8x^6 + 6x^5 + 2x^4\)[/tex], we can start by expanding each part step-by-step. Here is a detailed breakdown:
1. Expand the First Polynomial:
- First, consider the expression [tex]\(2x^4 \cdot (4x^2 + 3x + 1)\)[/tex].
- Distribute [tex]\(2x^4\)[/tex] across each term inside the parentheses:
- [tex]\(2x^4 \cdot 4x^2 = 8x^6\)[/tex]
- [tex]\(2x^4 \cdot 3x = 6x^5\)[/tex]
- [tex]\(2x^4 \cdot 1 = 2x^4\)[/tex]
The expanded form of the first polynomial is [tex]\(8x^6 + 6x^5 + 2x^4\)[/tex].
2. Multiply by the Second Polynomial:
- The expression [tex]\(8x^6 + 6x^5 + 2x^4\)[/tex] is already expanded.
3. Product of the Two Expanded Polynomials:
- Now, we need to multiply the single terms from both expanded forms obtained above:
[tex]\[
(8x^6 + 6x^5 + 2x^4) \times (8x^6 + 6x^5 + 2x^4)
\][/tex]
Multiplying each term in the first polynomial by each term in the second polynomial:
[tex]\[
\begin{align*}
8x^6 \cdot 8x^6 &= 64x^{12} \\
8x^6 \cdot 6x^5 &= 48x^{11} \\
8x^6 \cdot 2x^4 &= 16x^{10} \\
6x^5 \cdot 8x^6 &= 48x^{11} \\
6x^5 \cdot 6x^5 &= 36x^{10} \\
6x^5 \cdot 2x^4 &= 12x^9 \\
2x^4 \cdot 8x^6 &= 16x^{10} \\
2x^4 \cdot 6x^5 &= 12x^9 \\
2x^4 \cdot 2x^4 &= 4x^8 \\
\end{align*}
\][/tex]
4. Combine Like Terms:
- Combine terms with the same powers of [tex]\(x\)[/tex]:
[tex]\[
\begin{align*}
64x^{12} &\\
+ (48x^{11} + 48x^{11}) &= 96x^{11} \\
+ (16x^{10} + 36x^{10} + 16x^{10}) &= 68x^{10} \\
+ (12x^{9} + 12x^{9}) &= 24x^{9} \\
+ 4x^{8} & \\
\end{align*}
\][/tex]
5. Final Result:
- The polynomials combine to give the final product:
[tex]\[
64x^{12} + 96x^{11} + 68x^{10} + 24x^{9} + 4x^{8}
\][/tex]
This expression represents the product of the two given polynomials in expanded form.
1. Expand the First Polynomial:
- First, consider the expression [tex]\(2x^4 \cdot (4x^2 + 3x + 1)\)[/tex].
- Distribute [tex]\(2x^4\)[/tex] across each term inside the parentheses:
- [tex]\(2x^4 \cdot 4x^2 = 8x^6\)[/tex]
- [tex]\(2x^4 \cdot 3x = 6x^5\)[/tex]
- [tex]\(2x^4 \cdot 1 = 2x^4\)[/tex]
The expanded form of the first polynomial is [tex]\(8x^6 + 6x^5 + 2x^4\)[/tex].
2. Multiply by the Second Polynomial:
- The expression [tex]\(8x^6 + 6x^5 + 2x^4\)[/tex] is already expanded.
3. Product of the Two Expanded Polynomials:
- Now, we need to multiply the single terms from both expanded forms obtained above:
[tex]\[
(8x^6 + 6x^5 + 2x^4) \times (8x^6 + 6x^5 + 2x^4)
\][/tex]
Multiplying each term in the first polynomial by each term in the second polynomial:
[tex]\[
\begin{align*}
8x^6 \cdot 8x^6 &= 64x^{12} \\
8x^6 \cdot 6x^5 &= 48x^{11} \\
8x^6 \cdot 2x^4 &= 16x^{10} \\
6x^5 \cdot 8x^6 &= 48x^{11} \\
6x^5 \cdot 6x^5 &= 36x^{10} \\
6x^5 \cdot 2x^4 &= 12x^9 \\
2x^4 \cdot 8x^6 &= 16x^{10} \\
2x^4 \cdot 6x^5 &= 12x^9 \\
2x^4 \cdot 2x^4 &= 4x^8 \\
\end{align*}
\][/tex]
4. Combine Like Terms:
- Combine terms with the same powers of [tex]\(x\)[/tex]:
[tex]\[
\begin{align*}
64x^{12} &\\
+ (48x^{11} + 48x^{11}) &= 96x^{11} \\
+ (16x^{10} + 36x^{10} + 16x^{10}) &= 68x^{10} \\
+ (12x^{9} + 12x^{9}) &= 24x^{9} \\
+ 4x^{8} & \\
\end{align*}
\][/tex]
5. Final Result:
- The polynomials combine to give the final product:
[tex]\[
64x^{12} + 96x^{11} + 68x^{10} + 24x^{9} + 4x^{8}
\][/tex]
This expression represents the product of the two given polynomials in expanded form.
