Questions: Factor the expression that defines the function. (m(t)=t^3+7t^2-8t-56=)

Solution Steps

To factor the given cubic polynomial \( m(t) = t^3 + 7t^2 - 8t - 56 \), we can use the Rational Root Theorem to find potential rational roots. Once a root is found, we can perform polynomial division to factor the polynomial further.

We start with the polynomial given by the function: \[ m(t) = t^3 + 7t^2 - 8t - 56 \]

Using polynomial factorization techniques, we can express \( m(t) \) in a factored form. The polynomial can be factored as: \[ m(t) = (t + 7)(t^2 - 8) \]

The quadratic term \( t^2 - 8 \) can be further factored using the difference of squares: \[ t^2 - 8 = t^2 - 2\sqrt{2}^2 = (t - 2\sqrt{2})(t + 2\sqrt{2}) \] Thus, the complete factorization of \( m(t) \) is: \[ m(t) = (t + 7)(t - 2\sqrt{2})(t + 2\sqrt{2}) \]

Final Answer