Answer :
To multiply the polynomials [tex]\((8x^2 + 6x + 8)\)[/tex] and [tex]\((6x - 5)\)[/tex], we'll use the distributive property, often called the FOIL method when dealing with binomials. Since one of them is a trinomial, we'll distribute each term separately:
1. Distribute [tex]\(8x^2\)[/tex]:
[tex]\[
8x^2 \cdot 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \cdot (-5) = -40x^2
\][/tex]
2. Distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \cdot 6x = 36x^2
\][/tex]
[tex]\[
6x \cdot (-5) = -30x
\][/tex]
3. Distribute [tex]\(8\)[/tex]:
[tex]\[
8 \cdot 6x = 48x
\][/tex]
[tex]\[
8 \cdot (-5) = -40
\][/tex]
Now, we combine all of these results together:
[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]
Let's combine like terms:
- Cubic term:
[tex]\[
48x^3
\][/tex]
- Quadratic terms:
[tex]\[
(-40x^2 + 36x^2) = -4x^2
\][/tex]
- Linear terms:
[tex]\[
(-30x + 48x) = 18x
\][/tex]
- Constant term:
[tex]\[
-40
\][/tex]
Putting it all together, the resulting polynomial is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is D. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
1. Distribute [tex]\(8x^2\)[/tex]:
[tex]\[
8x^2 \cdot 6x = 48x^3
\][/tex]
[tex]\[
8x^2 \cdot (-5) = -40x^2
\][/tex]
2. Distribute [tex]\(6x\)[/tex]:
[tex]\[
6x \cdot 6x = 36x^2
\][/tex]
[tex]\[
6x \cdot (-5) = -30x
\][/tex]
3. Distribute [tex]\(8\)[/tex]:
[tex]\[
8 \cdot 6x = 48x
\][/tex]
[tex]\[
8 \cdot (-5) = -40
\][/tex]
Now, we combine all of these results together:
[tex]\[
48x^3 + (-40x^2) + 36x^2 + (-30x) + 48x + (-40)
\][/tex]
Let's combine like terms:
- Cubic term:
[tex]\[
48x^3
\][/tex]
- Quadratic terms:
[tex]\[
(-40x^2 + 36x^2) = -4x^2
\][/tex]
- Linear terms:
[tex]\[
(-30x + 48x) = 18x
\][/tex]
- Constant term:
[tex]\[
-40
\][/tex]
Putting it all together, the resulting polynomial is:
[tex]\[
48x^3 - 4x^2 + 18x - 40
\][/tex]
So, the correct answer is D. [tex]\(48x^3 - 4x^2 + 18x - 40\)[/tex].
