Solve 75x^3-5x^2-19x-3

$$75 { x }^{ 3 } -5 { x }^{ 2 } -19x-3$$

Factor

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

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 75. One such root is \frac{3}{5}. Factor the polynomial by dividing it by 5x-3.

$$\left(5x-3\right)\left(15x^{2}+8x+1\right)$$

Consider 15x^{2}+8x+1. Factor the expression by grouping. First, the expression needs to be rewritten as 15x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=8 ab=15\times 1=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite 15x^{2}+8x+1 as \left(15x^{2}+3x\right)+\left(5x+1\right).

$$\left(15x^{2}+3x\right)+\left(5x+1\right)$$

Factor out 3x in 15x^{2}+3x.

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

Factor out common term 5x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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