Question 10 Let \( f(x)=x^{3}+2 x^{2}-3 x-10 \). a) Find all zeros, both real and complex of \( f \). b) Write \( f \) in factored form, as the product of linear factors.

To find the zeros of the polynomial \( f(x)=x^{3}+2x^{2}-3x-10 \), we can start by applying the Rational Root Theorem, which tells us that any rational roots must be factors of the constant term (-10) over the leading coefficient (1). Testing possible rational roots like \( \pm 1, \pm 2, \pm 5, \pm 10 \), we find that \( x = 2 \) is a root since \( f(2) = 0 \). Next, we can perform synthetic division of \( f(x) \) by \( x - 2 \), which will give us a quadratic factor. The division results in \( x^2 + 4x + 5 \). We can find the remaining zeros by solving the quadratic equation \( x^2 + 4x + 5 = 0 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-4 \pm \sqrt{16 - 20}}{2} = \frac{-4 \pm \sqrt{-4}}{2} = -2 \pm i. \] Thus, the zeros of \( f(x) \) are \( 2, -2 + i, -2 - i \). To write \( f \) in factored form, we express it using its zeros: \[ f(x) = (x - 2)(x - (-2 + i))(x - (-2 - i)) = (x - 2)((x + 2) - i)((x + 2) + i). \] By recognizing the factored expression for the quadratic \( (x + 2)^2 + 1^2 \): Thus, the factored form is: \[ f(x) = (x - 2)\left((x + 2)^2 + 1\right). \] Now we have the complete picture of the polynomial’s behavior!