Answer :
To multiply the polynomials [tex]\((4x^2 + 3x + 7)\)[/tex] and [tex]\((8x - 5)\)[/tex], we'll follow these steps:
1. Distribute each term of the first polynomial by each term of the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by both terms in the second polynomial:
- [tex]\(4x^2 \times 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \times (-5) = -20x^2\)[/tex]
- Multiply [tex]\(3x\)[/tex] by both terms in the second polynomial:
- [tex]\(3x \times 8x = 24x^2\)[/tex]
- [tex]\(3x \times (-5) = -15x\)[/tex]
- Multiply [tex]\(7\)[/tex] by both terms in the second polynomial:
- [tex]\(7 \times 8x = 56x\)[/tex]
- [tex]\(7 \times (-5) = -35\)[/tex]
2. Combine all terms:
Combine the results from the distribution to form a single polynomial:
- [tex]\[32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) + (-35)\][/tex]
3. Simplify by combining like terms:
- [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
4. Final simplified polynomial:
So, after combining like terms, the polynomial is:
[tex]\[32x^3 + 4x^2 + 41x - 35\][/tex]
Therefore, the correct answer is option C: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
1. Distribute each term of the first polynomial by each term of the second polynomial:
- Multiply [tex]\(4x^2\)[/tex] by both terms in the second polynomial:
- [tex]\(4x^2 \times 8x = 32x^3\)[/tex]
- [tex]\(4x^2 \times (-5) = -20x^2\)[/tex]
- Multiply [tex]\(3x\)[/tex] by both terms in the second polynomial:
- [tex]\(3x \times 8x = 24x^2\)[/tex]
- [tex]\(3x \times (-5) = -15x\)[/tex]
- Multiply [tex]\(7\)[/tex] by both terms in the second polynomial:
- [tex]\(7 \times 8x = 56x\)[/tex]
- [tex]\(7 \times (-5) = -35\)[/tex]
2. Combine all terms:
Combine the results from the distribution to form a single polynomial:
- [tex]\[32x^3 + (-20x^2 + 24x^2) + (-15x + 56x) + (-35)\][/tex]
3. Simplify by combining like terms:
- [tex]\(x^2\)[/tex] terms: [tex]\(-20x^2 + 24x^2 = 4x^2\)[/tex]
- [tex]\(x\)[/tex] terms: [tex]\(-15x + 56x = 41x\)[/tex]
4. Final simplified polynomial:
So, after combining like terms, the polynomial is:
[tex]\[32x^3 + 4x^2 + 41x - 35\][/tex]
Therefore, the correct answer is option C: [tex]\(32x^3 + 4x^2 + 41x - 35\)[/tex].
