Answer :
Sure! Let's work through this problem of combining polynomials step-by-step. There are two expressions we need to simplify.
1. First expression:
We have:
[tex]\[
(7x^4 - 9x^3 - 8x) + (-x^4 + 9x^3 + 10)
\][/tex]
- Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(7x^4 + (-x^4)\)[/tex] gives [tex]\(6x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 + 9x^3\)[/tex] gives [tex]\(0x^3\)[/tex] (which is just 0, so we can omit it).
- For [tex]\(x\)[/tex]: [tex]\(-8x\)[/tex] has no like term to combine, so it remains [tex]\(-8x\)[/tex].
- The constant term: [tex]\(10\)[/tex].
So, the simplified form of the first expression is:
[tex]\[
6x^4 - 8x + 10
\][/tex]
2. Second expression:
We have:
[tex]\[
(-2x^4 + 9x^3 - 9x^2) - (-9x^4 - 9x^3 - 5x)
\][/tex]
- First, distribute the negative sign across the second polynomial:
[tex]\[
-2x^4 + 9x^3 - 9x^2 + 9x^4 + 9x^3 + 5x
\][/tex]
- Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(-2x^4 + 9x^4\)[/tex] gives [tex]\(7x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(9x^3 + 9x^3\)[/tex] gives [tex]\(18x^3\)[/tex].
- For [tex]\(x^2\)[/tex]: [tex]\(-9x^2\)[/tex] has no like term, so it remains [tex]\(-9x^2\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(5x\)[/tex].
Therefore, the simplified form of the second expression is:
[tex]\[
7x^4 + 18x^3 - 9x^2 + 5x
\][/tex]
Now, we've successfully simplified both polynomial expressions. The final results are:
- First expression: [tex]\(6x^4 - 8x + 10\)[/tex]
- Second expression: [tex]\(7x^4 + 18x^3 - 9x^2 + 5x\)[/tex]
1. First expression:
We have:
[tex]\[
(7x^4 - 9x^3 - 8x) + (-x^4 + 9x^3 + 10)
\][/tex]
- Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(7x^4 + (-x^4)\)[/tex] gives [tex]\(6x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(-9x^3 + 9x^3\)[/tex] gives [tex]\(0x^3\)[/tex] (which is just 0, so we can omit it).
- For [tex]\(x\)[/tex]: [tex]\(-8x\)[/tex] has no like term to combine, so it remains [tex]\(-8x\)[/tex].
- The constant term: [tex]\(10\)[/tex].
So, the simplified form of the first expression is:
[tex]\[
6x^4 - 8x + 10
\][/tex]
2. Second expression:
We have:
[tex]\[
(-2x^4 + 9x^3 - 9x^2) - (-9x^4 - 9x^3 - 5x)
\][/tex]
- First, distribute the negative sign across the second polynomial:
[tex]\[
-2x^4 + 9x^3 - 9x^2 + 9x^4 + 9x^3 + 5x
\][/tex]
- Combine like terms:
- For [tex]\(x^4\)[/tex]: [tex]\(-2x^4 + 9x^4\)[/tex] gives [tex]\(7x^4\)[/tex].
- For [tex]\(x^3\)[/tex]: [tex]\(9x^3 + 9x^3\)[/tex] gives [tex]\(18x^3\)[/tex].
- For [tex]\(x^2\)[/tex]: [tex]\(-9x^2\)[/tex] has no like term, so it remains [tex]\(-9x^2\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(5x\)[/tex].
Therefore, the simplified form of the second expression is:
[tex]\[
7x^4 + 18x^3 - 9x^2 + 5x
\][/tex]
Now, we've successfully simplified both polynomial expressions. The final results are:
- First expression: [tex]\(6x^4 - 8x + 10\)[/tex]
- Second expression: [tex]\(7x^4 + 18x^3 - 9x^2 + 5x\)[/tex]
