Questions: For the polynomial below, 3 is a zero. f(x) = x^3 + 3 x^2 - 12 x - 18 Express f(x) as a product of linear factors.

Solution Steps

To express the polynomial as a product of linear factors, we can use synthetic division to divide the polynomial by $(x-3)$ since 3 is a zero of the polynomial. The resulting quotient will be a quadratic polynomial, which can then be factored further.

Perform synthetic division by dividing \( f(x) = x^3 + 3x^2 - 12x - 18 \) by \( x - 3 \) since 3 is a zero of the polynomial: \[ \begin{array}{r|rrr} 3 & 1 & 3 & -12 & -18 \\ \hline & & 3 & 18 & 18 \\ \hline & 1 & 6 & 6 & 0 \\ \end{array} \] The quotient is \( x^2 + 6x + 6 \).

Factorize the quadratic quotient \( x^2 + 6x + 6 \) to express it as a product of linear factors: \[ x^2 + 6x + 6 = (x + 3)(x + 2) \]

Final Answer