Solve y^3-2y^2-29y-42

$$y ^ { 3 } - 2 y ^ { 2 } - 29 y - 42$$

Factor

$$\left(y-7\right)\left(y+2\right)\left(y+3\right)$$

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 -42 and q divides the leading coefficient 1. One such root is 7. Factor the polynomial by dividing it by y-7.

$$\left(y-7\right)\left(y^{2}+5y+6\right)$$

Consider y^{2}+5y+6. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+6. To find a and b, set up a system to be solved.

$$a+b=5 ab=1\times 6=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=2 b=3$$

Rewrite y^{2}+5y+6 as \left(y^{2}+2y\right)+\left(3y+6\right).

$$\left(y^{2}+2y\right)+\left(3y+6\right)$$

Factor out y in the first and 3 in the second group.

$$y\left(y+2\right)+3\left(y+2\right)$$

Factor out common term y+2 by using distributive property.

$$\left(y+2\right)\left(y+3\right)$$

Rewrite the complete factored expression.

$$\left(y-7\right)\left(y+2\right)\left(y+3\right)$$

Evaluate

$$\left(y-7\right)\left(y+2\right)\left(y+3\right)$$