Solve 4A^3-7A-3

$$4 A ^ { 3 } - 7 A - 3$$

Factor

$$\left(2A-3\right)\left(A+1\right)\left(2A+1\right)$$

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

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

Consider 2A^{2}+3A+1. Factor the expression by grouping. First, the expression needs to be rewritten as 2A^{2}+aA+bA+1. To find a and b, set up a system to be solved.

$$a+b=3 ab=2\times 1=2$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=2$$

Rewrite 2A^{2}+3A+1 as \left(2A^{2}+A\right)+\left(2A+1\right).

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

Factor out A in 2A^{2}+A.

$$A\left(2A+1\right)+2A+1$$

Factor out common term 2A+1 by using distributive property.

$$\left(2A+1\right)\left(A+1\right)$$

Rewrite the complete factored expression.

$$\left(2A-3\right)\left(A+1\right)\left(2A+1\right)$$

Evaluate

$$4A^{3}-7A-3$$