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Determine if [tex](y-1)[/tex] is a factor of [tex]y^3 - 9y^2 + 10y + 5[/tex].

A. Yes
B. Not enough information to determine
C. No
D. Only if [tex]x=-1[/tex]


Sagot :

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