Solve {l}{text{solutionstotheequation}y^3=27text{.Choo}}{text{allthatapply.}}{{llll}{-sqrt[3]{27}}&{sqrt[3]{27}}&{3}&{-3}}

$$\left. \begin{array} { l } { \text { solutions to the equation } y ^ { 3 } = 27 \text { . Choo } } \\ { \text { all that apply. } } \\ { \left. \begin{array} { l l l l } { - \sqrt[ 3 ] { 27 } } & { \sqrt[ 3 ] { 27 } } & { 3 } & { - 3 } \end{array} \right. } \end{array} \right.$$

Solve for y

$$y=3$$

Solution Steps

Subtract 27 from both sides.

$$y^{3}-27=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -27 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±27,±9,±3,±1$$

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.

$$y=3$$

By Factor theorem, y-k is a factor of the polynomial for each root k. Divide y^{3}-27 by y-3 to get y^{2}+3y+9. Solve the equation where the result equals to 0.

$$y^{2}+3y+9=0$$

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, 3 for b, and 9 for c in the quadratic formula.

$$y=\frac{-3±\sqrt{3^{2}-4\times 1\times 9}}{2}$$

Do the calculations.

$$y=\frac{-3±\sqrt{-27}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$y\in \emptyset$$

List all found solutions.

$$y=3$$