Solve 4b^4-3b^2+5b-6

$$4 b ^ { 4 } - 3 b ^ { 2 } + 5 b - 6$$

Factor

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

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 -6 and q divides the leading coefficient 4. One such root is -\frac{3}{2}. Factor the polynomial by dividing it by 2b+3.

$$\left(2b+3\right)\left(2b^{3}-3b^{2}+3b-2\right)$$

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

$$\left(b-1\right)\left(2b^{2}-b+2\right)$$

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

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

Evaluate

$$4b^{4}-3b^{2}+5b-6$$