Which polynomial is a factor of $x^3 - 5x^2 + 11x - 10$? A - HSC - SSCE Mathematics Extension 1 - Question 1 - 2017 - Paper 1

To determine which polynomial is a factor, we can use synthetic division or direct substitution to evaluate the polynomial at possible roots derived from the options provided.

Using synthetic division or substituting $x = 2$:

$P(2) = 2^3 - 5(2^2) + 11(2) - 10 = 8 - 20 + 22 - 10 = 0$

Since $P(2) = 0$, $x - 2$ is a factor.

Using synthetic division or substituting $x = -2$:

$P(-2) = (-2)^3 - 5(-2)^2 + 11(-2) - 10 = -8 - 20 - 22 - 10 = -60$

Since $P(-2) eq 0$, $x + 2$ is NOT a factor.

This expression is not a linear factor of the polynomial, and substituting any value will not yield a zero product. Hence, it is NOT a factor.

Using synthetic division or substitution methods may show this polynomial does not yield a factorization leading to a result of zero for the cubic polynomial.

The only polynomial that functions as a factor of $x^3 - 5x^2 + 11x - 10$ is A. $x - 2$.