1.6 Other Types of Equations

Example: Solve the polynomial equation

y^3+y^2=4y+4

Solution: Solve the polynomial equation by factoring.

The original equation.
Write the equation so that all the terms are on the same side. Subtract 4y Subtract 4
Group two terms pairs of terms and factor the greatest common factor from each group. The greatest common factor for the first group is The greatest common factor for the second group is -4
Factor the common binomial from each term. The common binomial is The other factor is formed using the coefficients of the parenthesis
Continue to factor completely by factoring the difference of squares in the second parenthesis
Apply the zero product property by setting each factor equal to zero.	or or
Solve each remaining equation.	or or or or or or

The solutions to the polynomial equation y^3+y^2=4y+4 are y=-1 or y=-2 or y=2 .

Example: Solve the equation.

$5x^{2/3}-6x^{1/3}+1=0$

Solution:

The equation is similar to a quadratic. It has 3 terms and one exponent is twice the other. Since the equation is quadratic in form, use substitution to solve the equation.

Use the following substitution to rewrite the equation

$u=x^{1/3}$

$u^2=x^{2/3}$

Original Equation	$5x^{2/3}-6x^{1/3}+1=0$
Substitute $u=x^{1/3}$ $u^2=x^{2/3}$	$5x^{2/3}-6x^{1/3}+1=0$ $5u^{2}-6u+1=0$
Solve the quadratic equation by factoring. 1) Factor the quadratic	$5u^{2}-6u+1=0$
Solve the quadratic equation by factoring. 2) Apply the zero product property	or
Solve the quadratic equation by factoring. 3) Solve each linear factor	or or or or or
Substitute again to bring back the original variable. Use the original substitution. $u=x^{1/3}$	or $x^{1/3}=1/5$ or $x^{1/3}=1$
Solve the equation with rational exponents. 1) Rewrite the rational exponents in radical form	$x^{1/3}=1/5$ or $x^{1/3}=1$ or
Solve the equation with rational exponents. 2) Cancel the cube root by cubing both sides. 3) Simplify	or $(root{3}{x})^3=(1/5)^3$ or $(root{3}{x})^3=(1)^3$ or