Problem 64 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ x^{3}-7 x^{2}+10 x=0 $$

Step-by-Step Solution

Answer

The solutions are x = 0, x = 2, x = 5.

Step 1: Factor the Equation

First, we look for common factors in each term of the equation. Notice that each term has an \( x \), so we can factor \( x \) out:\[ x(x^2 - 7x + 10) = 0 \].

Step 2: Solve for the First Solution

Since the equation is factored as \( x(x^2 - 7x + 10) = 0 \), we set each factor to zero. First, for the factor \( x \): \[ x = 0 \]. This is our first solution.

Step 3: Factor the Quadratic Expression

Now, focus on the quadratic factor \( x^2 - 7x + 10 \). We need to factor this expression. Look for two numbers that multiply to 10 and add up to -7. These numbers are -2 and -5, so:\[ x^2 - 7x + 10 = (x - 2)(x - 5) \].

Step 4: Solve for the Remaining Solutions

With the factored equation \( x(x - 2)(x - 5) = 0 \), set each factor equal to zero:1. \( x - 2 = 0 \Rightarrow x = 2 \).2. \( x - 5 = 0 \Rightarrow x = 5 \).These are the second and third solutions.

Key Concepts

Factoring

Factoring is a key technique in algebra used to simplify equations. When we factor an expression, we are essentially rewriting it as a product of simpler expressions. In the given problem, the equation is initially in the form of a cubic term. By observing each term, we notice that all terms share a common factor, which is \( x \). This observation allows us to factor \( x \) out of the equation, giving us \( x(x^2 - 7x + 10) = 0 \). This step simplifies the equation significantly and is crucial for solving the polynomial equation. Factoring transforms the problem into smaller, more manageable parts:

• Identify common factors that can be factored out.
• Try to recognize patterns like quadratic expressions that can be factored further.

Factoring is like unpacking a problem into smaller bits. Once factored, each part can be individually addressed, making the solving process easier.
It's essential for simplifying expressions and is a foundational skill in algebra.

Zero Product Property

The zero product property is a fundamental principle that states if a product of several factors is zero, at least one of the factors must be zero. This property is very useful in algebra when trying to find the roots of an equation. Once you have an equation in a factored form, like in our example \( x(x - 2)(x - 5) = 0 \), the zero product property allows us to set each factor to zero separately.This means you take each individual component:

• The first factor \( x \) gives us \( x = 0 \).
• The second factor \( x - 2 \) leads to \( x = 2 \).
• The third factor \( x - 5 \) results in \( x = 5 \).

By applying this property, you break down a complex problem into solvable pieces. Each value where a factor equals zero represents a solution to the original equation.
This method is particularly helpful in polynomial equations where the degree is higher, as breaking it down into its zero factors simplifies finding the solution.

Quadratic Expression

A quadratic expression is a polynomial of degree 2, generally written in the form \( ax^2 + bx + c \). In the example provided, the quadratic expression derived from factoring is \( x^2 - 7x + 10 \).To solve this, we need to factor it further, which involves finding two numbers that:

• Multiply to the constant term, in this case, 10.
• Add up to the linear coefficient, which is -7.

For our quadratic expression, the numbers -2 and -5 fit these conditions. Thus, the expression can be factored as \( (x - 2)(x - 5) \). Once factored, it can easily be used within the zero product property to find the solutions.Understanding how to deal with quadratic expressions is fundamental in algebra. It enables us to break down polynomial equations into solvable components. Recognizing patterns and relationships among coefficients and constant terms can simplify this process greatly.
Practice with this step solidifies one's ability to tackle various algebraic equations.