Answer :
Sure! Let's find the product of the two given expressions: [tex]\( (1 + 4x + 3x^2)(2 - 7x - 9x^2) \)[/tex].
To multiply these two polynomials, we use the distributive property (also known as the FOIL method for binomials) to ensure each term in the first polynomial multiplies each term in the second polynomial.
### Step-by-Step Multiplication:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\( 1 \times 2 = 2 \)[/tex]
- [tex]\( 1 \times (-7x) = -7x \)[/tex]
- [tex]\( 1 \times (-9x^2) = -9x^2 \)[/tex]
- [tex]\( 4x \times 2 = 8x \)[/tex]
- [tex]\( 4x \times (-7x) = -28x^2 \)[/tex]
- [tex]\( 4x \times (-9x^2) = -36x^3 \)[/tex]
- [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]
2. Combine all the terms obtained:
- [tex]\( 2 \)[/tex]
- [tex]\( -7x + 8x = x \)[/tex]
- [tex]\( -9x^2 - 28x^2 + 6x^2 = -31x^2 \)[/tex]
- [tex]\( -36x^3 - 21x^3 = -57x^3 \)[/tex]
- [tex]\( -27x^4 \)[/tex]
3. Write the final polynomial:
The product of the two expressions is:
[tex]\[ 2 + x - 31x^2 - 57x^3 - 27x^4 \][/tex]
Comparing this with the given options, we can see that option B matches our result. Therefore, the correct answer is:
B. [tex]\(2 + x - 31x^2 - 57x^3 - 27x^4\)[/tex]
To multiply these two polynomials, we use the distributive property (also known as the FOIL method for binomials) to ensure each term in the first polynomial multiplies each term in the second polynomial.
### Step-by-Step Multiplication:
1. Multiply each term in the first polynomial by each term in the second polynomial:
- [tex]\( 1 \times 2 = 2 \)[/tex]
- [tex]\( 1 \times (-7x) = -7x \)[/tex]
- [tex]\( 1 \times (-9x^2) = -9x^2 \)[/tex]
- [tex]\( 4x \times 2 = 8x \)[/tex]
- [tex]\( 4x \times (-7x) = -28x^2 \)[/tex]
- [tex]\( 4x \times (-9x^2) = -36x^3 \)[/tex]
- [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]
2. Combine all the terms obtained:
- [tex]\( 2 \)[/tex]
- [tex]\( -7x + 8x = x \)[/tex]
- [tex]\( -9x^2 - 28x^2 + 6x^2 = -31x^2 \)[/tex]
- [tex]\( -36x^3 - 21x^3 = -57x^3 \)[/tex]
- [tex]\( -27x^4 \)[/tex]
3. Write the final polynomial:
The product of the two expressions is:
[tex]\[ 2 + x - 31x^2 - 57x^3 - 27x^4 \][/tex]
Comparing this with the given options, we can see that option B matches our result. Therefore, the correct answer is:
B. [tex]\(2 + x - 31x^2 - 57x^3 - 27x^4\)[/tex]
