Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)\)[/tex] and [tex]\((6x - 5)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms.
Here is a step-by-step breakdown:
1. Multiply each term in [tex]\((8x^2 + 6x + 8)\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(6x \cdot 6x = 36x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(6x \cdot (-5) = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(8 \cdot 6x = 48x\)[/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(8 \cdot (-5) = -40\)[/tex]
2. Combine the results:
- Combine all the terms obtained from the multiplication:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\(-30x + 48x = 18x\)[/tex]
4. Write the final expression:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
The final polynomial after multiplying and combining like terms is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is:
A. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex]
Here is a step-by-step breakdown:
1. Multiply each term in [tex]\((8x^2 + 6x + 8)\)[/tex] by each term in [tex]\((6x - 5)\)[/tex]:
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(8x^2 \cdot 6x = 48x^3\)[/tex]
- Multiply [tex]\(8x^2\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(8x^2 \cdot (-5) = -40x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(6x \cdot 6x = 36x^2\)[/tex]
- Multiply [tex]\(6x\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(6x \cdot (-5) = -30x\)[/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(6x\)[/tex]:
[tex]\(8 \cdot 6x = 48x\)[/tex]
- Multiply [tex]\(8\)[/tex] by [tex]\(-5\)[/tex]:
[tex]\(8 \cdot (-5) = -40\)[/tex]
2. Combine the results:
- Combine all the terms obtained from the multiplication:
[tex]\[
48x^3 - 40x^2 + 36x^2 - 30x + 48x - 40
\][/tex]
3. Combine like terms:
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\(-40x^2 + 36x^2 = -4x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\(-30x + 48x = 18x\)[/tex]
4. Write the final expression:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
The final polynomial after multiplying and combining like terms is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is:
A. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex]
