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Answer :
Sure, let's work through the multiplication of the polynomials step-by-step:
Given the polynomials:
[tex]\[
(5x^2 + 2x + 8)(7x - 6)
\][/tex]
We need to multiply each term in the first polynomial by each term in the second polynomial. This is achieved through distribution.
### Step-by-Step Distribution:
1. Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply [tex]\(2x\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply [tex]\(8\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
### Combine All the Results:
Now, combine all the terms obtained:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x - 48
\][/tex]
### Simplify by Combining Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
So, the polynomial after simplification is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
The final answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
Given the polynomials:
[tex]\[
(5x^2 + 2x + 8)(7x - 6)
\][/tex]
We need to multiply each term in the first polynomial by each term in the second polynomial. This is achieved through distribution.
### Step-by-Step Distribution:
1. Multiply [tex]\(5x^2\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply [tex]\(2x\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply [tex]\(8\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
### Combine All the Results:
Now, combine all the terms obtained:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x - 48
\][/tex]
### Simplify by Combining Like Terms:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine the [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
So, the polynomial after simplification is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
The final answer is:
A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex]
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