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Answer :
Sure, let's go through the process step-by-step to multiply the polynomials [tex]\((4x^2 + 3x + 7)(8x - 5)\)[/tex].
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, distribute the [tex]\(4x^2\)[/tex] term:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \times (-5) = -20x^2
\][/tex]
- Next, distribute the [tex]\(3x\)[/tex] term:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
[tex]\[
3x \times (-5) = -15x
\][/tex]
- Finally, distribute the [tex]\(7\)[/tex] constant term:
[tex]\[
7 \times 8x = 56x
\][/tex]
[tex]\[
7 \times (-5) = -35
\][/tex]
2. Combine all the distributed terms:
[tex]\[
32x^3 - 20x^2 + 24x^2 - 15x + 56x - 35
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-15x + 56x = 41x\)[/tex]
4. Write the final polynomial:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
So, the correctly multiplied polynomial is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
1. Distribute each term in the first polynomial to each term in the second polynomial:
- First, distribute the [tex]\(4x^2\)[/tex] term:
[tex]\[
4x^2 \times 8x = 32x^3
\][/tex]
[tex]\[
4x^2 \times (-5) = -20x^2
\][/tex]
- Next, distribute the [tex]\(3x\)[/tex] term:
[tex]\[
3x \times 8x = 24x^2
\][/tex]
[tex]\[
3x \times (-5) = -15x
\][/tex]
- Finally, distribute the [tex]\(7\)[/tex] constant term:
[tex]\[
7 \times 8x = 56x
\][/tex]
[tex]\[
7 \times (-5) = -35
\][/tex]
2. Combine all the distributed terms:
[tex]\[
32x^3 - 20x^2 + 24x^2 - 15x + 56x - 35
\][/tex]
3. Combine like terms:
- For [tex]\(x^2\)[/tex]: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- For [tex]\(x\)[/tex]: [tex]\(-15x + 56x = 41x\)[/tex]
4. Write the final polynomial:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
So, the correctly multiplied polynomial is:
[tex]\[
32x^3 + 4x^2 + 41x - 35
\][/tex]
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