Solve 10x^3-33x^2+8x+3

$$10 x ^ { 3 } - 33 x ^ { 2 } + 8 x + 3$$

Factor

$$\left(x-3\right)\left(2x-1\right)\left(5x+1\right)$$

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 10. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1.

$$\left(2x-1\right)\left(5x^{2}-14x-3\right)$$

Consider 5x^{2}-14x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-14 ab=5\left(-3\right)=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -14.

$$a=-15 b=1$$

Rewrite 5x^{2}-14x-3 as \left(5x^{2}-15x\right)+\left(x-3\right).

$$\left(5x^{2}-15x\right)+\left(x-3\right)$$

Factor out 5x in 5x^{2}-15x.

$$5x\left(x-3\right)+x-3$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(5x+1\right)$$

Rewrite the complete factored expression.

$$\left(x-3\right)\left(2x-1\right)\left(5x+1\right)$$

Evaluate

$$\left(x-3\right)\left(2x-1\right)\left(5x+1\right)$$