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Answer :
To solve the problem, we need to find the sum of two polynomial functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], which are given as:
1. [tex]\( f(x) = -7x^5 + 9x^7 + 1 + 3x^6 \)[/tex]
2. [tex]\( g(x) = 2 - 5x^6 + 7x^7 - 2x^4 \)[/tex]
### Step-by-Step Solution:
1. Find [tex]\((f + g)(x)\)[/tex]:
To find the sum [tex]\((f + g)(x)\)[/tex], we combine like terms from both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = -7x^5 + 9x^7 + 1 + 3x^6
\][/tex]
[tex]\[
g(x) = 2 - 5x^6 + 7x^7 - 2x^4
\][/tex]
We add these polynomials by combining the coefficients of the same powers of [tex]\( x \)[/tex]:
- [tex]\( x^7 \)[/tex] term: [tex]\( 9x^7 + 7x^7 = 16x^7 \)[/tex]
- [tex]\( x^6 \)[/tex] term: [tex]\( 3x^6 - 5x^6 = -2x^6 \)[/tex]
- [tex]\( x^5 \)[/tex] term: Only in [tex]\( f(x) \)[/tex], so it's [tex]\(-7x^5\)[/tex]
- [tex]\( x^4 \)[/tex] term: Only in [tex]\( g(x) \)[/tex], so it's [tex]\(-2x^4\)[/tex]
- Constant term: [tex]\( 1 + 2 = 3 \)[/tex]
Combining all, we get:
[tex]\[
(f + g)(x) = 16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]
2. Evaluate [tex]\((f + g)(1)\)[/tex]:
Now, substitute [tex]\( x = 1 \)[/tex] into [tex]\((f + g)(x)\)[/tex]:
[tex]\[
(f + g)(1) = 16(1)^7 - 2(1)^6 - 7(1)^5 - 2(1)^4 + 3
\][/tex]
Simplify this expression:
- [tex]\( 16(1) - 2(1) - 7(1) - 2(1) + 3 = 16 - 2 - 7 - 2 + 3 \)[/tex]
- Combine the terms: [tex]\( 16 - 2 - 7 - 2 + 3 = 8 \)[/tex]
Therefore, [tex]\((f + g)(1) = 8\)[/tex].
So the final polynomial [tex]\((f + g)(x)\)[/tex] is:
[tex]\[
16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]
And [tex]\((f + g)(1)\)[/tex] evaluates to 8.
1. [tex]\( f(x) = -7x^5 + 9x^7 + 1 + 3x^6 \)[/tex]
2. [tex]\( g(x) = 2 - 5x^6 + 7x^7 - 2x^4 \)[/tex]
### Step-by-Step Solution:
1. Find [tex]\((f + g)(x)\)[/tex]:
To find the sum [tex]\((f + g)(x)\)[/tex], we combine like terms from both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = -7x^5 + 9x^7 + 1 + 3x^6
\][/tex]
[tex]\[
g(x) = 2 - 5x^6 + 7x^7 - 2x^4
\][/tex]
We add these polynomials by combining the coefficients of the same powers of [tex]\( x \)[/tex]:
- [tex]\( x^7 \)[/tex] term: [tex]\( 9x^7 + 7x^7 = 16x^7 \)[/tex]
- [tex]\( x^6 \)[/tex] term: [tex]\( 3x^6 - 5x^6 = -2x^6 \)[/tex]
- [tex]\( x^5 \)[/tex] term: Only in [tex]\( f(x) \)[/tex], so it's [tex]\(-7x^5\)[/tex]
- [tex]\( x^4 \)[/tex] term: Only in [tex]\( g(x) \)[/tex], so it's [tex]\(-2x^4\)[/tex]
- Constant term: [tex]\( 1 + 2 = 3 \)[/tex]
Combining all, we get:
[tex]\[
(f + g)(x) = 16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]
2. Evaluate [tex]\((f + g)(1)\)[/tex]:
Now, substitute [tex]\( x = 1 \)[/tex] into [tex]\((f + g)(x)\)[/tex]:
[tex]\[
(f + g)(1) = 16(1)^7 - 2(1)^6 - 7(1)^5 - 2(1)^4 + 3
\][/tex]
Simplify this expression:
- [tex]\( 16(1) - 2(1) - 7(1) - 2(1) + 3 = 16 - 2 - 7 - 2 + 3 \)[/tex]
- Combine the terms: [tex]\( 16 - 2 - 7 - 2 + 3 = 8 \)[/tex]
Therefore, [tex]\((f + g)(1) = 8\)[/tex].
So the final polynomial [tex]\((f + g)(x)\)[/tex] is:
[tex]\[
16x^7 - 2x^6 - 7x^5 - 2x^4 + 3
\][/tex]
And [tex]\((f + g)(1)\)[/tex] evaluates to 8.
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