Question

factor f(x) into linear factors given that k is a zero of f(x).
f(x) = 3x³ + 4x² - 69x + 90; k = 3
f(x) = \square (factor completely.)

Explanation:

Step1: Use Factor Theorem

Since \( k = 3 \) is a zero of \( f(x) \), \( (x - 3) \) is a factor of \( f(x) \). We perform polynomial long division or use synthetic division to divide \( f(x)=3x^{3}+4x^{2}-69x + 90 \) by \( (x - 3) \). Let's use synthetic division:
The coefficients of \( f(x) \) are \( 3,4,-69,90 \) and we divide by \( x - 3 \) (so we use root \( 3 \)).
\[

$$\begin{array}{r|rrrr} 3 & 3 & 4 & -69 & 90 \\ & & 9 & 39 & -90 \\ \hline & 3 & 13 & -30 & 0 \\ \end{array}$$

\]
So the quotient polynomial is \( 3x^{2}+13x - 30 \). Thus, \( f(x)=(x - 3)(3x^{2}+13x - 30) \).

Step2: Factor the quadratic

We factor \( 3x^{2}+13x - 30 \). We need two numbers \( a \) and \( b \) such that \( a\times b=3\times(-30)=-90 \) and \( a + b = 13 \). The numbers are \( 18 \) and \( - 5 \).
\[

$$\begin{align*} 3x^{2}+13x - 30&=3x^{2}+18x-5x - 30\\ &=3x(x + 6)-5(x + 6)\\ &=(3x - 5)(x + 6) \end{align*}$$

\]

Step3: Write the linear factors

Combining the factors, we have \( f(x)=(x - 3)(3x - 5)(x + 6) \).

Answer:

\( (x - 3)(3x - 5)(x + 6) \)