Questions: For the polynomial below, 2 is a zero. h(x) = x^3 - 6x + 4 Express h(x) as a product of linear factors.

Transcript text: For the polynomial below, 2 is a zero. \[ h(x)=x^{3}-6 x+4 \] Express $h(x)$ as a product of linear factors.

Solution

Solution Steps

Step 1: Reducing the cubic polynomial to a quadratic polynomial

Given the cubic polynomial $h(x) = x^3 + 0x^2 - 6x + 4$ and a known zero $x = 2$, we perform synthetic division to obtain a quadratic polynomial $q(x) = x^2 + 2x - 2$.

Step 2: Solving the quadratic equation to find the other two zeros

The discriminant of the quadratic equation is $\Delta = 12$, indicating real roots. Using the quadratic formula, the roots are calculated as $x = 0.732$ and $x = -2.732$.

Step 3: Expressing the original cubic polynomial as a product of its linear factors

The original cubic polynomial can be expressed as $h(x) = 1(x - (2))(x - (0.732))(x - (-2.732))$.