Problem 20 - Contemporary Precalculus

Question

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{1 / 3}+11 x^{1 / 6}+24$$

Step-by-Step Solution

Answer

Answer: The factored form of the expression $$x^{1/3} + 11x^{1/6} + 24$$ is $$(x^{1/6} + 8)(x^{1/6} + 3)$$.

Step 1: Substitute the Variable

Let's make a substitution of the variable: Let $$y = x^{1/6}$$, so we have $$y^2 = x^{1/3}$$. Now rewrite the given expression in terms of y: $$y^2 + 11y + 24$$.

Step 2: Factor the Quadratic Expression

Now we have a quadratic expression in terms of y: $$y^2 + 11y + 24$$. We can factor this easily. We are looking for two factors that multiply to 24 and add up to 11. The two numbers are 8 and 3. So, we can rewrite the quadratic as: $$(y + 8)(y + 3)$$.

Step 3: Resubstitute the Variable

Now we need to convert back to x. Since we made the substitution $$y = x^{1/6}$$, we can write our factored expression as: $$(x^{1/6} + 8)(x^{1/6} + 3)$$. The factored form of the given expression $$x^{1/3} + 11x^{1/6} + 24$$ is $$(x^{1/6} + 8)(x^{1/6} + 3)$$.

Key Concepts

Variable Substitution

Variable substitution is a powerful algebraic tool, especially when dealing with complex expressions. It simplifies an expression or an equation by replacing a part of it with a new variable. This technique often transforms a challenging problem into a more manageable one.

For example, considering the expression with exponents such as exponents and radicals, \(x^{1/3} + 11x^{1/6} + 24\), it can seem intimidating at first. But by introducing a substitute \(y = x^{1/6}\), we convert the expression into the quadratic form \(y^2 + 11y + 24\).

Substitution doesn't alter the mathematical meaning; it's a strategic maneuver. After solving the simpler equation, you 'resubstitute' to revert to the original variable. It's like translating a sentence into another language to understand it better and then translating it back.

Factoring Quadratic Expressions

Factoring quadratic expressions is a foundational skill in algebra. A quadratic expression is typically in the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. To factor such an expression means to write it as the product of two binomials.

Here's a simplified approach:

• Identify \(a\), \(b\), and \(c\) in the standard form of the quadratic equation.
• Find two numbers that multiply to \(ac\) and add up to \(b\).
• Use these two numbers to break up the middle term \(bx\) and factor by grouping, or write them directly into two binomials.

In the exercise, \(y^2 + 11y + 24\) is factored into \(y + 8\) and \(y + 3\), which are the two binomials whose product gives the original quadratic expression.

Exponents and Radicals

Exponents and radicals are expressions that represent repeated multiplication. An exponent, such as \(x^2\), indicates that \(x\) is to be multiplied by itself. A radical, such as \(\sqrt[3]{x}\), indicates the root of a number, which is the inverse operation of raising a number to a power.

In the context of our problem, the expression deals with fractional exponents like \(x^{1/3}\) and \(x^{1/6}\), which can also be written as \(\sqrt[3]{x}\) and \(\sqrt[6]{x}\). Understanding how to manipulate these forms is crucial in simplifying and factoring the expression. The exercise shows that knowledge of how exponents work makes it easier to perform variable substitution and factor the expression back into terms of the original variable.