Solve P(x)=4x^3+8x^2+5x+1

$$P ( x ) = 4 x ^ { 3 } + 8 x ^ { 2 } + 5 x + 1$$

Factor

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

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

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

Consider 2x^{2}+3x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+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 2x^{2}+3x+1 as \left(2x^{2}+x\right)+\left(2x+1\right).

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

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

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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