Given that [tex]\( y + 1 \)[/tex] is a factor of the polynomial [tex]\( k y^3 + y^2 - 2 y + 4 y - 10 \)[/tex], we can determine the value of [tex]\( k \)[/tex] by following these steps:
1. Combine Like Terms in the Polynomial:
[tex]\[
k y^3 + y^2 - 2 y + 4 y - 10
\][/tex]
Combine the [tex]\( y \)[/tex]-terms:
[tex]\[
k y^3 + y^2 + 2 y - 10
\][/tex]
So the polynomial simplifies to:
[tex]\[
k y^3 + y^2 + 2 y - 10
\][/tex]
2. Factor Theorem Application:
According to the factor theorem, if [tex]\( y + 1 \)[/tex] is a factor of the polynomial, substituting [tex]\( y = -1 \)[/tex] into the polynomial should yield 0.
3. Substitute [tex]\( y = -1 \)[/tex] into the Polynomial:
[tex]\[
k(-1)^3 + (-1)^2 + 2(-1) - 10
\][/tex]
Simplify the expression:
[tex]\[
k(-1) + 1 - 2 - 10
\][/tex]
[tex]\[
-k + 1 - 2 - 10 = 0
\][/tex]
[tex]\[
-k - 11 = 0
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
Isolate [tex]\( k \)[/tex] in the equation [tex]\( -k - 11 = 0 \)[/tex]:
[tex]\[
-k - 11 = 0
\][/tex]
[tex]\[
-k = 11
\][/tex]
[tex]\[
k = -11
\][/tex]
Thus, the value of [tex]\( k \)[/tex] is:
[tex]\[
k = -11
\][/tex]
