Solve -16t^4+24t^3-9t^2+1

$$-16 { t }^{ 4 } +24 { t }^{ 3 } -9 { t }^{ 2 } +1$$

Factor

$$\left(t-1\right)\left(4t+1\right)\left(-4t^{2}+3t-1\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient -16. One such root is -\frac{1}{4}. Factor the polynomial by dividing it by 4t+1.

$$\left(4t+1\right)\left(-4t^{3}+7t^{2}-4t+1\right)$$

Consider -4t^{3}+7t^{2}-4t+1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient -4. One such root is 1. Factor the polynomial by dividing it by t-1.

$$\left(t-1\right)\left(-4t^{2}+3t-1\right)$$

Rewrite the complete factored expression. Polynomial -4t^{2}+3t-1 is not factored since it does not have any rational roots.

$$\left(t-1\right)\left(-4t^{2}+3t-1\right)\left(4t+1\right)$$

Evaluate

$$1-9t^{2}+24t^{3}-16t^{4}$$