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Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). Let's break it down step-by-step:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- 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]
- 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]
- Multiply [tex]\(8\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
2. Combine all the products:
- [tex]\(35x^3\)[/tex]
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex] (Combine the [tex]\(x^2\)[/tex] terms)
- [tex]\(-12x + 56x = 44x\)[/tex] (Combine the [tex]\(x\)[/tex] terms)
- [tex]\(-48\)[/tex]
3. Combine all these terms to form the final polynomial:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the multiplied result of the polynomials is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Therefore, the correct answer is option D: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
1. Multiply each term in the first polynomial by each term in the second polynomial:
- 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]
- 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]
- Multiply [tex]\(8\)[/tex] by each term in [tex]\(7x - 6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
2. Combine all the products:
- [tex]\(35x^3\)[/tex]
- [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex] (Combine the [tex]\(x^2\)[/tex] terms)
- [tex]\(-12x + 56x = 44x\)[/tex] (Combine the [tex]\(x\)[/tex] terms)
- [tex]\(-48\)[/tex]
3. Combine all these terms to form the final polynomial:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the multiplied result of the polynomials is:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
Therefore, the correct answer is option D: [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
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