Answer :
Sure! Here is a step-by-step solution to multiply the trinomials [tex]\((x^2 + 8x - 5)\)[/tex] and [tex]\((3x^2 - x + 8)\)[/tex]:
1. Distribute each term in the first trinomial to each term in the second trinomial:
We need to multiply each term of [tex]\((x^2 + 8x - 5)\)[/tex] by each term of [tex]\((3x^2 - x + 8)\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(x^2 \cdot 3x^2 = 3x^4\)[/tex]
- [tex]\(x^2 \cdot (-x) = -x^3\)[/tex]
- [tex]\(x^2 \cdot 8 = 8x^2\)[/tex]
3. Multiply [tex]\(8x\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(8x \cdot 3x^2 = 24x^3\)[/tex]
- [tex]\(8x \cdot (-x) = -8x^2\)[/tex]
- [tex]\(8x \cdot 8 = 64x\)[/tex]
4. Multiply [tex]\(-5\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(-5 \cdot 3x^2 = -15x^2\)[/tex]
- [tex]\(-5 \cdot (-x) = 5x\)[/tex]
- [tex]\(-5 \cdot 8 = -40\)[/tex]
5. Add all the resulting terms together:
[tex]\[
3x^4 + (-x^3) + 8x^2 + 24x^3 + (-8x^2) + 64x + (-15x^2) + 5x + (-40)
\][/tex]
6. Combine like terms:
- For [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-x^3 + 24x^3 = 23x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 - 8x^2 - 15x^2 = -15x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(64x + 5x = 69x\)[/tex]
- Constant term: [tex]\(-40\)[/tex]
So, the final polynomial after multiplying the trinomials [tex]\((x^2 + 8x - 5)\)[/tex] and [tex]\((3x^2 - x + 8)\)[/tex] is:
[tex]\[
3x^4 + 23x^3 - 15x^2 + 69x - 40
\][/tex]
Thus, the correct answer is:
[tex]\[
3x^4 + 23x^3 - 15x^2 + 69x - 40
\][/tex]
1. Distribute each term in the first trinomial to each term in the second trinomial:
We need to multiply each term of [tex]\((x^2 + 8x - 5)\)[/tex] by each term of [tex]\((3x^2 - x + 8)\)[/tex].
2. Multiply [tex]\(x^2\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(x^2 \cdot 3x^2 = 3x^4\)[/tex]
- [tex]\(x^2 \cdot (-x) = -x^3\)[/tex]
- [tex]\(x^2 \cdot 8 = 8x^2\)[/tex]
3. Multiply [tex]\(8x\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(8x \cdot 3x^2 = 24x^3\)[/tex]
- [tex]\(8x \cdot (-x) = -8x^2\)[/tex]
- [tex]\(8x \cdot 8 = 64x\)[/tex]
4. Multiply [tex]\(-5\)[/tex] with each term in [tex]\((3x^2 - x + 8)\)[/tex]:
- [tex]\(-5 \cdot 3x^2 = -15x^2\)[/tex]
- [tex]\(-5 \cdot (-x) = 5x\)[/tex]
- [tex]\(-5 \cdot 8 = -40\)[/tex]
5. Add all the resulting terms together:
[tex]\[
3x^4 + (-x^3) + 8x^2 + 24x^3 + (-8x^2) + 64x + (-15x^2) + 5x + (-40)
\][/tex]
6. Combine like terms:
- For [tex]\(x^4\)[/tex] terms: [tex]\(3x^4\)[/tex]
- For [tex]\(x^3\)[/tex] terms: [tex]\(-x^3 + 24x^3 = 23x^3\)[/tex]
- For [tex]\(x^2\)[/tex] terms: [tex]\(8x^2 - 8x^2 - 15x^2 = -15x^2\)[/tex]
- For [tex]\(x\)[/tex] terms: [tex]\(64x + 5x = 69x\)[/tex]
- Constant term: [tex]\(-40\)[/tex]
So, the final polynomial after multiplying the trinomials [tex]\((x^2 + 8x - 5)\)[/tex] and [tex]\((3x^2 - x + 8)\)[/tex] is:
[tex]\[
3x^4 + 23x^3 - 15x^2 + 69x - 40
\][/tex]
Thus, the correct answer is:
[tex]\[
3x^4 + 23x^3 - 15x^2 + 69x - 40
\][/tex]
