To simplify the polynomial expression [tex]\(\left(5 v^2+4 v+6\right)+\left(-4 v^2+8 v-2\right)-\left(9 v^2-8 v-6\right)\)[/tex], follow these steps:
1. Identify and group similar terms:
- Identify the quadratic ([tex]\(v^2\)[/tex]), linear ([tex]\(v\)[/tex]), and constant terms in each polynomial.
- For [tex]\(5 v^2 + 4 v + 6\)[/tex]:
- [tex]\(5v^2\)[/tex] (quadratic term)
- [tex]\(4v\)[/tex] (linear term)
- [tex]\(6\)[/tex] (constant term)
- For [tex]\(-4 v^2 + 8 v - 2\)[/tex]:
- [tex]\(-4v^2\)[/tex] (quadratic term)
- [tex]\(8v\)[/tex] (linear term)
- [tex]\(-2\)[/tex] (constant term)
- For [tex]\(9 v^2 - 8 v - 6\)[/tex]:
- [tex]\(9v^2\)[/tex] (quadratic term)
- [tex]\(-8v\)[/tex] (linear term)
- [tex]\(-6\)[/tex] (constant term)
2. Combine the quadratic terms:
- Sum of the quadratic terms: [tex]\(5v^2 + (-4v^2) - 9v^2\)[/tex].
- First, combine [tex]\(5v^2\)[/tex] and [tex]\(-4v^2\)[/tex]:
[tex]\[
5v^2 - 4v^2 = 1v^2
\][/tex]
- Then, subtract [tex]\(9v^2\)[/tex]:
[tex]\[
1v^2 - 9v^2 = -8v^2
\][/tex]
- The resulting quadratic term is [tex]\(-8v^2\)[/tex].
3. Combine the linear terms:
- Sum of the linear terms: [tex]\(4v + 8v - (-8v)\)[/tex].
- First, add [tex]\(4v\)[/tex] and [tex]\(8v\)[/tex]:
[tex]\[
4v + 8v = 12v
\][/tex]
- Then, add [tex]\(8v\)[/tex] (since [tex]\(-(-8v)\)[/tex] is [tex]\(+8v\)[/tex]):
[tex]\[
12v + 8v = 20v
\][/tex]
- The resulting linear term is [tex]\(20v\)[/tex].
4. Combine the constant terms:
- Sum of the constant terms: [tex]\(6 + (-2) - (-6)\)[/tex].
- First, combine [tex]\(6\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
6 - 2 = 4
\][/tex]
- Then, add [tex]\(6\)[/tex] (since [tex]\(-(-6)\)[/tex] is [tex]\(+6\)[/tex]):
[tex]\[
4 + 6 = 10
\][/tex]
- The resulting constant term is [tex]\(10\)[/tex].
Therefore, the simplified polynomial expression is:
[tex]\[
-8v^2 + 20v + 10
\][/tex]
