Solve 3x^3-23x^2+13x+7

$$3 x ^ { 3 } - 23 x ^ { 2 } + 13 x + 7$$

Factor

$$\left(x-7\right)\left(x-1\right)\left(3x+1\right)$$

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 7 and q divides the leading coefficient 3. One such root is -\frac{1}{3}. Factor the polynomial by dividing it by 3x+1.

$$\left(3x+1\right)\left(x^{2}-8x+7\right)$$

Consider x^{2}-8x+7. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\times 7=7$$

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

$$a=-7 b=-1$$

Rewrite x^{2}-8x+7 as \left(x^{2}-7x\right)+\left(-x+7\right).

$$\left(x^{2}-7x\right)+\left(-x+7\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-7\right)-\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(x-1\right)$$

Rewrite the complete factored expression.

$$\left(x-7\right)\left(x-1\right)\left(3x+1\right)$$

Evaluate

$$\left(x-7\right)\left(x-1\right)\left(3x+1\right)$$