Answer :
To find the product of the polynomials [tex]\((1 + 4x + 3x^2)\)[/tex] and [tex]\((2 - 7x - 9x^2)\)[/tex], we'll follow the method of polynomial multiplication. Here's a step-by-step solution:
1. Distribute each term in the first polynomial by every term in the second polynomial. This is like applying the distributive property multiple times.
2. Multiply terms systematically:
- Start with the constant term in the first polynomial:
[tex]\[
1 \times 2 = 2
\][/tex]
[tex]\[
1 \times (-7x) = -7x
\][/tex]
[tex]\[
1 \times (-9x^2) = -9x^2
\][/tex]
- Next, the [tex]\(4x\)[/tex] term in the first polynomial:
[tex]\[
4x \times 2 = 8x
\][/tex]
[tex]\[
4x \times (-7x) = -28x^2
\][/tex]
[tex]\[
4x \times (-9x^2) = -36x^3
\][/tex]
- Lastly, the [tex]\(3x^2\)[/tex] term in the first polynomial:
[tex]\[
3x^2 \times 2 = 6x^2
\][/tex]
[tex]\[
3x^2 \times (-7x) = -21x^3
\][/tex]
[tex]\[
3x^2 \times (-9x^2) = -27x^4
\][/tex]
3. Combine like terms:
- The constant term: [tex]\(2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-7x + 8x = x\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 - 28x^2 + 6x^2 = -31x^2\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-36x^3 - 21x^3 = -57x^3\)[/tex]
- The highest degree term: [tex]\(-27x^4\)[/tex]
4. Form the final expression by assembling all the combined terms:
[tex]\[
2 + x - 31x^2 - 57x^3 - 27x^4
\][/tex]
Therefore, the product of the polynomials [tex]\((1 + 4x + 3x^2)\)[/tex] and [tex]\((2 - 7x - 9x^2)\)[/tex] is:
[tex]\[ 2 + x - 31x^2 - 57x^3 - 27x^4 \][/tex]
The correct answer is option B: [tex]\(2 + x - 31x^2 - 57x^3 - 27x^4\)[/tex].
1. Distribute each term in the first polynomial by every term in the second polynomial. This is like applying the distributive property multiple times.
2. Multiply terms systematically:
- Start with the constant term in the first polynomial:
[tex]\[
1 \times 2 = 2
\][/tex]
[tex]\[
1 \times (-7x) = -7x
\][/tex]
[tex]\[
1 \times (-9x^2) = -9x^2
\][/tex]
- Next, the [tex]\(4x\)[/tex] term in the first polynomial:
[tex]\[
4x \times 2 = 8x
\][/tex]
[tex]\[
4x \times (-7x) = -28x^2
\][/tex]
[tex]\[
4x \times (-9x^2) = -36x^3
\][/tex]
- Lastly, the [tex]\(3x^2\)[/tex] term in the first polynomial:
[tex]\[
3x^2 \times 2 = 6x^2
\][/tex]
[tex]\[
3x^2 \times (-7x) = -21x^3
\][/tex]
[tex]\[
3x^2 \times (-9x^2) = -27x^4
\][/tex]
3. Combine like terms:
- The constant term: [tex]\(2\)[/tex]
- The [tex]\(x\)[/tex] terms: [tex]\(-7x + 8x = x\)[/tex]
- The [tex]\(x^2\)[/tex] terms: [tex]\(-9x^2 - 28x^2 + 6x^2 = -31x^2\)[/tex]
- The [tex]\(x^3\)[/tex] terms: [tex]\(-36x^3 - 21x^3 = -57x^3\)[/tex]
- The highest degree term: [tex]\(-27x^4\)[/tex]
4. Form the final expression by assembling all the combined terms:
[tex]\[
2 + x - 31x^2 - 57x^3 - 27x^4
\][/tex]
Therefore, the product of the polynomials [tex]\((1 + 4x + 3x^2)\)[/tex] and [tex]\((2 - 7x - 9x^2)\)[/tex] is:
[tex]\[ 2 + x - 31x^2 - 57x^3 - 27x^4 \][/tex]
The correct answer is option B: [tex]\(2 + x - 31x^2 - 57x^3 - 27x^4\)[/tex].
