In Exercises 101–106, solve each equation. x(x + 1)^3 - 42(x + 1)^2 = 0

Verified step by step guidance

1. Factor out the common term $(x+1{)}^2$ from the equation: $(x+1{)}^2[x(x+1)-42]=0$.
2. Set each factor equal to zero: $(x+1{)}^2=0$ and $x(x+1)-42=0$.
3. Solve the first equation $(x+1{)}^2=0$ by taking the square root of both sides, which gives $x+1=0$.
4. Solve the second equation $x(x+1)-42=0$ by expanding and rearranging it to form a quadratic equation: $x^2+x-42=0$.
5. Use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ to solve the quadratic equation $x^2+x-42=0$, where $a=1$, $b=1$, and $c=-42$.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In the context of polynomial equations, factoring can simplify the equation, making it easier to solve for the variable. Recognizing common factors and applying techniques such as grouping or using the difference of squares are essential skills in algebra.

Zero Product Property

The Zero Product Property states that if the product of two or more factors equals zero, at least one of the factors must be zero. This principle is crucial when solving polynomial equations, as it allows us to set each factor equal to zero and solve for the variable. Understanding this property helps in finding all possible solutions to the equation.

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole number exponents, combined using addition, subtraction, and multiplication. They can have one or more terms and can be solved using various methods, including factoring, graphing, or applying the quadratic formula. Recognizing the degree and behavior of polynomial functions is important for understanding their solutions.