Answer :
To multiply the polynomials [tex]\((5x^2 + 2x + 8)(7x - 6)\)[/tex], we'll follow a systematic approach known as the distributive property or the FOIL method for binomials, although here we apply it to polynomials with more than two terms. Here's the step-by-step solution:
1. Distribute each term of the first polynomial to each term of the second polynomial:
[tex]\((5x^2)\)[/tex] is multiplied by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Next, distribute [tex]\((2x)\)[/tex] by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Finally, distribute [tex]\((8)\)[/tex] by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
4. Combine all the terms we obtained:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
5. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
6. Write down the final polynomial:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the answer is A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
1. Distribute each term of the first polynomial to each term of the second polynomial:
[tex]\((5x^2)\)[/tex] is multiplied by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times -6 = -30x^2\)[/tex]
2. Next, distribute [tex]\((2x)\)[/tex] by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times -6 = -12x\)[/tex]
3. Finally, distribute [tex]\((8)\)[/tex] by both [tex]\(7x\)[/tex] and [tex]\(-6\)[/tex]:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times -6 = -48\)[/tex]
4. Combine all the terms we obtained:
[tex]\[
35x^3 + (-30x^2) + 14x^2 + (-12x) + 56x + (-48)
\][/tex]
5. Combine like terms:
- Combine [tex]\(x^2\)[/tex] terms: [tex]\(-30x^2 + 14x^2 = -16x^2\)[/tex]
- Combine [tex]\(x\)[/tex] terms: [tex]\(-12x + 56x = 44x\)[/tex]
6. Write down the final polynomial:
[tex]\[
35x^3 - 16x^2 + 44x - 48
\][/tex]
So, the answer is A. [tex]\(35x^3 - 16x^2 + 44x - 48\)[/tex].
