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Answer :
To multiply the two polynomials [tex]\((8x + 9)\)[/tex] and [tex]\((3x^2 + x - 1)\)[/tex], we will use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. Here's a step-by-step breakdown:
1. Multiply [tex]\(8x\)[/tex] by each term in the second polynomial:
- [tex]\(8x \times 3x^2 = 24x^3\)[/tex]
- [tex]\(8x \times x = 8x^2\)[/tex]
- [tex]\(8x \times (-1) = -8x\)[/tex]
2. Multiply [tex]\(9\)[/tex] by each term in the second polynomial:
- [tex]\(9 \times 3x^2 = 27x^2\)[/tex]
- [tex]\(9 \times x = 9x\)[/tex]
- [tex]\(9 \times (-1) = -9\)[/tex]
3. Combine all the terms:
- Now we combine all the results from steps 1 and 2:
[tex]\[
24x^3 + 8x^2 - 8x + 27x^2 + 9x - 9
\][/tex]
4. Combine like terms:
- For the [tex]\(x^3\)[/tex] term: [tex]\(24x^3\)[/tex] (no other [tex]\(x^3\)[/tex] terms to combine with)
- For the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 27x^2 = 35x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-8x + 9x = 1x\)[/tex]
- Constant term: [tex]\(-9\)[/tex]
5. Write the final expression:
- Combining all the like terms from step 4, we get:
[tex]\[
24x^3 + 35x^2 + 1x - 9
\][/tex]
So, the product of [tex]\((8x + 9)\)[/tex] and [tex]\((3x^2 + x - 1)\)[/tex] is [tex]\(24x^3 + 35x^2 + x - 9\)[/tex].
1. Multiply [tex]\(8x\)[/tex] by each term in the second polynomial:
- [tex]\(8x \times 3x^2 = 24x^3\)[/tex]
- [tex]\(8x \times x = 8x^2\)[/tex]
- [tex]\(8x \times (-1) = -8x\)[/tex]
2. Multiply [tex]\(9\)[/tex] by each term in the second polynomial:
- [tex]\(9 \times 3x^2 = 27x^2\)[/tex]
- [tex]\(9 \times x = 9x\)[/tex]
- [tex]\(9 \times (-1) = -9\)[/tex]
3. Combine all the terms:
- Now we combine all the results from steps 1 and 2:
[tex]\[
24x^3 + 8x^2 - 8x + 27x^2 + 9x - 9
\][/tex]
4. Combine like terms:
- For the [tex]\(x^3\)[/tex] term: [tex]\(24x^3\)[/tex] (no other [tex]\(x^3\)[/tex] terms to combine with)
- For the [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 + 27x^2 = 35x^2\)[/tex]
- For the [tex]\(x\)[/tex] terms: [tex]\(-8x + 9x = 1x\)[/tex]
- Constant term: [tex]\(-9\)[/tex]
5. Write the final expression:
- Combining all the like terms from step 4, we get:
[tex]\[
24x^3 + 35x^2 + 1x - 9
\][/tex]
So, the product of [tex]\((8x + 9)\)[/tex] and [tex]\((3x^2 + x - 1)\)[/tex] is [tex]\(24x^3 + 35x^2 + x - 9\)[/tex].
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