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Answer :
To address the problem of determining if a given value [tex]\( k \)[/tex] is a zero of a polynomial, we need to check whether substituting [tex]\( k \)[/tex] into the polynomial results in zero. If it does, then [tex]\( k \)[/tex] is a zero of the polynomial. If not, we find the value of the polynomial at [tex]\( k \)[/tex].
Let's go step-by-step for the given polynomial:
1. Understand the Polynomial:
The polynomial given is:
[tex]\[
f(x) = x^5 - 23x^4 - 95x^3 + 70x^2 + 204x - 72
\][/tex]
We need to check if [tex]\( k = 1 \)[/tex] is a zero of this polynomial.
2. Substitute [tex]\( k = 1 \)[/tex] into the Polynomial:
Plug [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[
f(1) = 1^5 - 23 \times 1^4 - 95 \times 1^3 + 70 \times 1^2 + 204 \times 1 - 72
\][/tex]
3. Calculate Each Term:
Simplify step-by-step:
- [tex]\( 1^5 = 1 \)[/tex]
- [tex]\( -23 \times 1^4 = -23 \)[/tex]
- [tex]\( -95 \times 1^3 = -95 \)[/tex]
- [tex]\( 70 \times 1^2 = 70 \)[/tex]
- [tex]\( 204 \times 1 = 204 \)[/tex]
- Constant term = [tex]\(-72\)[/tex]
Now, plug in these values:
[tex]\[
f(1) = 1 - 23 - 95 + 70 + 204 - 72
\][/tex]
4. Combine the Terms:
[tex]\[
f(1) = 1 - 23 - 95 + 70 + 204 - 72 = 85
\][/tex]
5. Determine if [tex]\( k = 1 \)[/tex] is a Zero:
Since [tex]\( f(1) = 85 \)[/tex], which is not zero, [tex]\( k = 1 \)[/tex] is not a zero of the polynomial. Instead, [tex]\( f(1) = 85 \)[/tex].
Therefore, the solution tells us that [tex]\( k = 1 \)[/tex] is not a zero of the polynomial, and the value of the polynomial at [tex]\( k = 1 \)[/tex] is 85.
Let's go step-by-step for the given polynomial:
1. Understand the Polynomial:
The polynomial given is:
[tex]\[
f(x) = x^5 - 23x^4 - 95x^3 + 70x^2 + 204x - 72
\][/tex]
We need to check if [tex]\( k = 1 \)[/tex] is a zero of this polynomial.
2. Substitute [tex]\( k = 1 \)[/tex] into the Polynomial:
Plug [tex]\( x = 1 \)[/tex] into the polynomial:
[tex]\[
f(1) = 1^5 - 23 \times 1^4 - 95 \times 1^3 + 70 \times 1^2 + 204 \times 1 - 72
\][/tex]
3. Calculate Each Term:
Simplify step-by-step:
- [tex]\( 1^5 = 1 \)[/tex]
- [tex]\( -23 \times 1^4 = -23 \)[/tex]
- [tex]\( -95 \times 1^3 = -95 \)[/tex]
- [tex]\( 70 \times 1^2 = 70 \)[/tex]
- [tex]\( 204 \times 1 = 204 \)[/tex]
- Constant term = [tex]\(-72\)[/tex]
Now, plug in these values:
[tex]\[
f(1) = 1 - 23 - 95 + 70 + 204 - 72
\][/tex]
4. Combine the Terms:
[tex]\[
f(1) = 1 - 23 - 95 + 70 + 204 - 72 = 85
\][/tex]
5. Determine if [tex]\( k = 1 \)[/tex] is a Zero:
Since [tex]\( f(1) = 85 \)[/tex], which is not zero, [tex]\( k = 1 \)[/tex] is not a zero of the polynomial. Instead, [tex]\( f(1) = 85 \)[/tex].
Therefore, the solution tells us that [tex]\( k = 1 \)[/tex] is not a zero of the polynomial, and the value of the polynomial at [tex]\( k = 1 \)[/tex] is 85.
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