Questions: Factor f(x) = 3x^3 + 2x^2 - 7x + 2 into linear factors given that -2 is a zero of f(x).

Transcript text: Factor $f(x)=3 x^{3}+2 x^{2}-7 x+2$ into linear factors given that -2 is a zero of $f(x)$.

Solution

Solution Steps

To factor the polynomial $ f(x) = 3x^3 + 2x^2 - 7x + 2 $ completely, given that -2 is a zero, we can use synthetic division to divide the polynomial by $ x + 2 $. This will give us a quadratic polynomial, which we can then factor further to find the remaining linear factors.

Step 1: Synthetic Division

We start with the polynomial $ f(x) = 3x^3 + 2x^2 - 7x + 2 $ and perform synthetic division by $ x + 2 $ since -2 is a known zero. The coefficients of the polynomial are $ [3, 2, -7, 2] $. After performing the synthetic division, we obtain the new coefficients for the resulting quadratic polynomial, which are $ [3, -4, 1] $.

Step 2: Finding Roots of the Quadratic

Next, we need to find the roots of the quadratic polynomial $ 3x^2 - 4x + 1 $. Using the quadratic formula, we find the roots to be: \[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm 2}{6} \] This simplifies to: \[ x = 1 \quad \text{and} \quad x = \frac{1}{3} \]

Step 3: Writing the Linear Factors

Now that we have the roots, we can express the polynomial $ f(x) $ as a product of its linear factors. The linear factors corresponding to the roots are: \[ (x - 1) \quad \text{and} \quad \left(x - \frac{1}{3}\right) \] Including the factor from the zero $ -2 $, we have: \[ f(x) = 3(x + 2)(x - 1)\left(x - \frac{1}{3}\right) \]