Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)(6x - 5)\)[/tex], we will use the distributive property (also known as the FOIL method when applicable) to expand the expression.
Here's the step-by-step solution:
1. Distribute each term of the first polynomial to each term of the second polynomial:
- First, distribute [tex]\(8x^2\)[/tex]:
[tex]\[
8x^2 \times 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \times (-5) = -40x^2
\][/tex]
- Next, distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \times 6x = 36x^2
\][/tex]
[tex]\[
6x \times (-5) = -30x
\][/tex]
- Finally, distribute [tex]\(8\)[/tex]:
[tex]\[
8 \times 6x = 48x
\][/tex]
[tex]\[
8 \times (-5) = -40
\][/tex]
2. Combine all the terms:
[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-40x^2 + 36x^2 = -4x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 48x = 18x
\][/tex]
- The constant term remains [tex]\(-40\)[/tex].
4. Write the final expanded polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is Option B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
Here's the step-by-step solution:
1. Distribute each term of the first polynomial to each term of the second polynomial:
- First, distribute [tex]\(8x^2\)[/tex]:
[tex]\[
8x^2 \times 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \times (-5) = -40x^2
\][/tex]
- Next, distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \times 6x = 36x^2
\][/tex]
[tex]\[
6x \times (-5) = -30x
\][/tex]
- Finally, distribute [tex]\(8\)[/tex]:
[tex]\[
8 \times 6x = 48x
\][/tex]
[tex]\[
8 \times (-5) = -40
\][/tex]
2. Combine all the terms:
[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]
3. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-40x^2 + 36x^2 = -4x^2
\][/tex]
- Combine [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 48x = 18x
\][/tex]
- The constant term remains [tex]\(-40\)[/tex].
4. Write the final expanded polynomial:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is Option B: [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
