Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
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 \( \boxed{2} \).
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
