Solve 4a^3+4a^2-7a+2

$$4 a ^ { 3 } + 4 a ^ { 2 } - 7 a + 2$$

Factor

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

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 2 and q divides the leading coefficient 4. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2a-1.

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

Consider 2a^{2}+3a-2. Factor the expression by grouping. First, the expression needs to be rewritten as 2a^{2}+pa+qa-2. To find p and q, set up a system to be solved.

$$p+q=3 pq=2\left(-2\right)=-4$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$p=-1 q=4$$

Rewrite 2a^{2}+3a-2 as \left(2a^{2}-a\right)+\left(4a-2\right).

$$\left(2a^{2}-a\right)+\left(4a-2\right)$$

Factor out a in the first and 2 in the second group.

$$a\left(2a-1\right)+2\left(2a-1\right)$$

Factor out common term 2a-1 by using distributive property.

$$\left(2a-1\right)\left(a+2\right)$$

Rewrite the complete factored expression.

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

Evaluate

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