Solve 10x^3-49x^2-8x+15

$$10 { x }^{ 3 } -49 { x }^{ 2 } -8x+15$$

Factor

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

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 15 and q divides the leading coefficient 10. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1.

$$\left(2x-1\right)\left(5x^{2}-22x-15\right)$$

Consider 5x^{2}-22x-15. Factor the expression by grouping. First, the expression needs to be rewritten as 5x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-22 ab=5\left(-15\right)=-75$$

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 -75.

$$1,-75 3,-25 5,-15$$

Calculate the sum for each pair.

$$1-75=-74 3-25=-22 5-15=-10$$

The solution is the pair that gives sum -22.

$$a=-25 b=3$$

Rewrite 5x^{2}-22x-15 as \left(5x^{2}-25x\right)+\left(3x-15\right).

$$\left(5x^{2}-25x\right)+\left(3x-15\right)$$

Factor out 5x in the first and 3 in the second group.

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

Factor out common term x-5 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

$$10x^{3}-49x^{2}-8x+15$$