To determine if [tex]\((y-1)\)[/tex] is a factor of the polynomial [tex]\(y^3 - 9y^2 + 10y + 5\)[/tex], we can use the Factor Theorem. The Factor Theorem states that [tex]\((y-c)\)[/tex] is a factor of a polynomial [tex]\(P(y)\)[/tex] if and only if [tex]\(P(c) = 0\)[/tex].
Here, we need to check if [tex]\((y-1)\)[/tex] is a factor, which means we need to determine if [tex]\(P(1) = 0\)[/tex] for the polynomial [tex]\(P(y) = y^3 - 9y^2 + 10y + 5\)[/tex].
1. Substitute [tex]\(y = 1\)[/tex] into the polynomial:
[tex]\[ P(1) = (1)^3 - 9(1)^2 + 10(1) + 5 \][/tex]
2. Calculate each term:
[tex]\[ (1)^3 = 1 \][/tex]
[tex]\[ -9(1)^2 = -9 \][/tex]
[tex]\[ 10(1) = 10 \][/tex]
[tex]\[ 5 \text{ is a constant term} \][/tex]
3. Sum the calculated values:
[tex]\[ P(1) = 1 - 9 + 10 + 5 \][/tex]
4. Perform the arithmetic:
[tex]\[ P(1) = 1 - 9 + 10 + 5 = 7 \][/tex]
Since [tex]\(P(1) = 7\)[/tex] is not equal to 0, [tex]\((y-1)\)[/tex] is not a factor of the polynomial [tex]\(y^3 - 9y^2 + 10y + 5\)[/tex].
Therefore, the correct answer is:
C. No
