Answer :
Sure! Let's multiply the polynomials [tex]\((5x^2 + 2x + 8)\)[/tex] and [tex]\((7x - 6)\)[/tex] step by step.
We will use the distributive property, which means we'll multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the first term of the first polynomial by each term of the second polynomial:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply the third term of the first polynomial by each term of the second polynomial:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, let's combine all these results:
[tex]\[35x^3\][/tex] (from the first step)
[tex]\[-30x^2 + 14x^2 = -16x^2\][/tex] (combining the [tex]\(x^2\)[/tex] terms from the first and second steps)
[tex]\[-12x + 56x = 44x\][/tex] (combining the [tex]\(x\)[/tex] terms from the second and third steps)
The constant term is [tex]\(-48\)[/tex].
Putting it all together, we get the polynomial:
[tex]\[35x^3 - 16x^2 + 44x - 48\][/tex]
From the given options, this corresponds to option C.
We will use the distributive property, which means we'll multiply each term in the first polynomial by each term in the second polynomial.
1. Multiply the first term of the first polynomial by each term of the second polynomial:
- [tex]\(5x^2 \times 7x = 35x^3\)[/tex]
- [tex]\(5x^2 \times (-6) = -30x^2\)[/tex]
2. Multiply the second term of the first polynomial by each term of the second polynomial:
- [tex]\(2x \times 7x = 14x^2\)[/tex]
- [tex]\(2x \times (-6) = -12x\)[/tex]
3. Multiply the third term of the first polynomial by each term of the second polynomial:
- [tex]\(8 \times 7x = 56x\)[/tex]
- [tex]\(8 \times (-6) = -48\)[/tex]
Now, let's combine all these results:
[tex]\[35x^3\][/tex] (from the first step)
[tex]\[-30x^2 + 14x^2 = -16x^2\][/tex] (combining the [tex]\(x^2\)[/tex] terms from the first and second steps)
[tex]\[-12x + 56x = 44x\][/tex] (combining the [tex]\(x\)[/tex] terms from the second and third steps)
The constant term is [tex]\(-48\)[/tex].
Putting it all together, we get the polynomial:
[tex]\[35x^3 - 16x^2 + 44x - 48\][/tex]
From the given options, this corresponds to option C.
