IDNLearn.com offers a user-friendly platform for finding and sharing answers. Get prompt and accurate answers to your questions from our community of knowledgeable experts.
Sagot :
Let's find the value of [tex]\( k \)[/tex] such that [tex]\( x + k \)[/tex] is a factor of the polynomial [tex]\( x^2 + kx + k - 2 \)[/tex].
### Step-by-Step Solution:
1. Factor Theorem Application:
According to the Factor Theorem, [tex]\( x + k \)[/tex] is a factor of the polynomial [tex]\( x^2 + kx + k - 2 \)[/tex] if and only if substituting [tex]\( x = -k \)[/tex] into the polynomial yields zero.
2. Substitution and Setting the Polynomial to Zero:
Substitute [tex]\( x = -k \)[/tex] into the polynomial:
[tex]\[ (-k)^2 + k(-k) + k - 2 = 0 \][/tex]
3. Simplify the Substitution:
Simplify the expression:
[tex]\[ k^2 - k^2 + k - 2 = 0 \][/tex]
[tex]\[ 0 + k - 2 = 0 \][/tex]
[tex]\[ k - 2 = 0 \][/tex]
4. Solve for [tex]\( k \)[/tex]:
Solving the equation [tex]\( k - 2 = 0 \)[/tex], we find:
[tex]\[ k = 2 \][/tex]
Therefore, the value of [tex]\( k \)[/tex] for which [tex]\( x + k \)[/tex] is a factor of [tex]\( x^2 + k x + k - 2 \)[/tex] is [tex]\( k = 2 \)[/tex].
### Step-by-Step Solution:
1. Factor Theorem Application:
According to the Factor Theorem, [tex]\( x + k \)[/tex] is a factor of the polynomial [tex]\( x^2 + kx + k - 2 \)[/tex] if and only if substituting [tex]\( x = -k \)[/tex] into the polynomial yields zero.
2. Substitution and Setting the Polynomial to Zero:
Substitute [tex]\( x = -k \)[/tex] into the polynomial:
[tex]\[ (-k)^2 + k(-k) + k - 2 = 0 \][/tex]
3. Simplify the Substitution:
Simplify the expression:
[tex]\[ k^2 - k^2 + k - 2 = 0 \][/tex]
[tex]\[ 0 + k - 2 = 0 \][/tex]
[tex]\[ k - 2 = 0 \][/tex]
4. Solve for [tex]\( k \)[/tex]:
Solving the equation [tex]\( k - 2 = 0 \)[/tex], we find:
[tex]\[ k = 2 \][/tex]
Therefore, the value of [tex]\( k \)[/tex] for which [tex]\( x + k \)[/tex] is a factor of [tex]\( x^2 + k x + k - 2 \)[/tex] is [tex]\( k = 2 \)[/tex].
Answer:
2
Step-by-step explanation:
To determine the value of \( k \) for which \( x + k \) is a factor of \( x^2 + kx + k - 2 \), we proceed as follows:
Since \( x + k \) is a factor, the polynomial \( x^2 + kx + k - 2 \) must be divisible by \( x + k \). By the factor theorem, if \( x + k \) is a factor, then \( x = -k \) is a root of the polynomial.
Substitute \( x = -k \) into \( x^2 + kx + k - 2 \):
\[
(-k)^2 + k(-k) + k - 2 = k^2 - k^2 + k - 2 = k - 2
\]
For \( x + k \) to be a factor, \( k - 2 \) must equal \( 0 \) (since the remainder when \( x^2 + kx + k - 2 \) is divided by \( x + k \) should be zero):
\[
k - 2 = 0
\]
\[
k = 2
\]
Therefore, the value of \( k \) for which \( x + k \) is a factor of \( x^2 + kx + k - 2 \) is \( \boxed{2} \).
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
