Solve left(y-1right)left(y+5right)left(y+2right)=00

$$\left( y-1 \right) \left( y+5 \right) \left( y+2 \right) =00$$

Solve for y

$$y=-5$$

$$y=1$$

$$y=-2$$

Solution Steps

Use the distributive property to multiply y-1 by y+5 and combine like terms.

$$\left(y^{2}+4y-5\right)\left(y+2\right)=0\times 0$$

Use the distributive property to multiply y^{2}+4y-5 by y+2 and combine like terms.

$$y^{3}+6y^{2}+3y-10=0\times 0$$

Multiply 0 and 0 to get 0.

$$y^{3}+6y^{2}+3y-10=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -10 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±10,±5,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$y=1$$

By Factor theorem, y-k is a factor of the polynomial for each root k. Divide y^{3}+6y^{2}+3y-10 by y-1 to get y^{2}+7y+10. Solve the equation where the result equals to 0.

$$y^{2}+7y+10=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 7 for b, and 10 for c in the quadratic formula.

$$y=\frac{-7±\sqrt{7^{2}-4\times 1\times 10}}{2}$$

Do the calculations.

$$y=\frac{-7±3}{2}$$

Solve the equation y^{2}+7y+10=0 when ± is plus and when ± is minus.

$$y=-5 y=-2$$

List all found solutions.

$$y=1 y=-5 y=-2$$