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Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we use the distributive property, which involves multiplying each term in the first polynomial by each term in the second polynomial. Here's a step-by-step breakdown:
1. Multiply the first term of the first polynomial with each term of the second polynomial:
[tex]\[
8x^2 \cdot 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \cdot (-5) = -40x^2
\][/tex]
2. Multiply the second term of the first polynomial with each term of the second polynomial:
[tex]\[
6x \cdot 6x = 36x^2
\][/tex]
[tex]\[
6x \cdot (-5) = -30x
\][/tex]
3. Multiply the third term of the first polynomial with each term of the second polynomial:
[tex]\[
8 \cdot 6x = 48x
\][/tex]
[tex]\[
8 \cdot (-5) = -40
\][/tex]
4. Combine all the terms:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
5. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 48x = 18x\)[/tex]
6. Write the final polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the product of the polynomials is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
The correct answer is:
C. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex]
1. Multiply the first term of the first polynomial with each term of the second polynomial:
[tex]\[
8x^2 \cdot 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \cdot (-5) = -40x^2
\][/tex]
2. Multiply the second term of the first polynomial with each term of the second polynomial:
[tex]\[
6x \cdot 6x = 36x^2
\][/tex]
[tex]\[
6x \cdot (-5) = -30x
\][/tex]
3. Multiply the third term of the first polynomial with each term of the second polynomial:
[tex]\[
8 \cdot 6x = 48x
\][/tex]
[tex]\[
8 \cdot (-5) = -40
\][/tex]
4. Combine all the terms:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
5. Combine like terms:
- The [tex]\(x^2\)[/tex] terms: [tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-30x + 48x = 18x\)[/tex]
6. Write the final polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the product of the polynomials is [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
The correct answer is:
C. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex]
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