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Answer :
To multiply the polynomials [tex]\((7x^2 + 5x + 7)(4x - 6)\)[/tex], we will use the distributive property. This means we will distribute each term in the first polynomial to every term in the second polynomial and then combine like terms. Let's go through this step by step.
1. Distribute 7x²:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times -6 = -42x^2
\][/tex]
2. Distribute 5x:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times -6 = -30x
\][/tex]
3. Distribute 7:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times -6 = -42
\][/tex]
4. Combine like terms:
Now, we need to add together all the terms:
[tex]\[
28x^3 - 42x^2 + 20x^2 - 30x + 28x - 42
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
Putting it all together, the resulting polynomial is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So the answer is B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
1. Distribute 7x²:
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7x^2 \times 4x = 28x^3
\][/tex]
- Multiply [tex]\(7x^2\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7x^2 \times -6 = -42x^2
\][/tex]
2. Distribute 5x:
- Multiply [tex]\(5x\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
5x \times 4x = 20x^2
\][/tex]
- Multiply [tex]\(5x\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
5x \times -6 = -30x
\][/tex]
3. Distribute 7:
- Multiply [tex]\(7\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
7 \times 4x = 28x
\][/tex]
- Multiply [tex]\(7\)[/tex] by [tex]\(-6\)[/tex]:
[tex]\[
7 \times -6 = -42
\][/tex]
4. Combine like terms:
Now, we need to add together all the terms:
[tex]\[
28x^3 - 42x^2 + 20x^2 - 30x + 28x - 42
\][/tex]
- Combine the [tex]\(x^2\)[/tex] terms:
[tex]\[
-42x^2 + 20x^2 = -22x^2
\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[
-30x + 28x = -2x
\][/tex]
Putting it all together, the resulting polynomial is:
[tex]\[
28x^3 - 22x^2 - 2x - 42
\][/tex]
So the answer is B. [tex]\(28x^3 - 22x^2 - 2x - 42\)[/tex].
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