Problem 57 - Algebra A Combined Function

Question

Solve each equation. $$ x^{3}-12 x^{2}+32 x=0 $$

Step-by-Step Solution

Answer

Factor the equation as \( x(x-4)(x-8)=0 \).

Step 1: Factor out the GCF

\(x^3 - 12x^2 + 32x = x(x^2 - 12x + 32) = 0\)

Step 2: Factor the quadratic

Find two numbers that multiply to 32 and add to \(-12\): \(-4\) and \(-8\).
\(x^2 - 12x + 32 = (x-4)(x-8)\)

Step 3: Solve

\(x(x-4)(x-8) = 0\)
\(x = 0\), \(x = 4\), or \(x = 8\)

Key Concepts

common factor

When tackling polynomial equations, identifying a common factor can significantly simplify the problem. A common factor is a term or number that divides each term in the polynomial without leaving a remainder. In the polynomial equation \(x^3 - 12x^2 + 32x = 0\), the common factor of all terms is \(x\). This is because each term in the polynomial has at least one \(x\) in it.

To factor out the common factor, you remove \(x\) from each term, effectively dividing each term by \(x\). This process simplifies the equation, resulting in:

• First Term: \(x^3 \div x = x^2\)
• Second Term: \(-12x^2 \div x = -12x\)
• Third Term: \(32x \div x = 32\)

The newly factored equation becomes \(x(x^2 - 12x + 32) = 0\). This step is crucial because it reduces the equation's complexity and paves the way for further factorization or finding the roots.

cubic equations

Cubic equations are polynomial equations of the third degree, meaning the highest power of the variable \(x\) is 3. The general form of a cubic equation is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\). These equations can often be more challenging to solve than quadratic equations due to their higher degree.

However, when a cubic equation has a common factor, as seen in \(x^3 - 12x^2 + 32x = 0\), this advantage can be used to break the equation down. Once the common factor \(x\) is factored out, the equation simplifies to a quadratic equation \(x^2 - 12x + 32 = 0\). Solving this quadratic equation will help in finding some or all roots of the original cubic equation. The presence of a cubic equation indicates there are up to three real roots. Factoring and solving each part separately after factoring out common terms often helps to find these roots efficiently.

For the given equation, factoring is a key step because it reduces the polynomial to a lower degree, making it solvable using methods suitable for quadratics.

factoring quadratic equations

Factoring quadratic equations is a method used to solve polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Once a cubic equation is reduced to a quadratic equation, it is often easier to solve.

The equation \(x^2 - 12x + 32 = 0\) is a quadratic equation that emerged after factoring the common \(x\) from the original cubic equation. To factor a quadratic equation, you need to find two numbers that multiply to the constant term \(c\) (in this case, \(32\)) and add up to the linear coefficient \(b\) (\(-12\)).

Looking at the equation:

• The two numbers that multiply to \(32\) and add to \(-12\) are \(-4\) and \(-8\).

This allows us to express the quadratic equation as \((x - 4)(x - 8) = 0\). Solving this gives the values \(x = 4\) and \(x = 8\). Those are the roots of the quadratic equation and, therefore, part of the solution to the cubic equation. Solving each factor set to zero helps determine all possible solutions for \(x\), including the solutions from the common factor part, which is \(x = 0\). This gives the complete set of solutions for the original equation as \(x = 0\), \(x = 4\), and \(x = 8\).