Question

If \(\frac{(C+x)}{x-3}=\frac{x+8}{3}\), which of the following could be an expression of \(C\) in terms of \(x\) ? A) \(3(1+x)\) B) \(x^2+2 x-24\) C) \(\frac{1}{3}(x+6)(x-4)\) D) \(\frac{1}{3}(x-3)(x+8)\)

Short answer

The correct answer is B) \(x^2+2x-24\).

Step-by-step solution

Write down the given equation

The given equation is: \[\frac{(C+x)}{x-3}=\frac{x+8}{3}\]

Cross-multiply the equation

Cross-multiply to eliminate the fractions: \[(C+x)(3)=(x-3)(x+8)\]

Expand both sides of the equation

Expand the equation: \[3C+3x=(x^2-3x+8x-24)\]

Simplify the equation

Simplify the equation by combining like terms: \[3C+3x=x^2+5x-24\]

Isolate the variable, C

In order to find the expression for C, isolate it on one side of the equation: \[3C=x^2+5x-24-3x\] Now simplify the equation: \[3C=x^2+2x-24\]

Match the expression with one of the answer choices

We have found the expression for C: \[3C=x^2+2x-24\] Now, compare this expression with the given answer choices. We can see that the expression matches answer choice B: \(x^2+2x-24\). So, the correct answer is B) \(x^2+2x-24\).

Key concepts

Cross-multiplication in Algebraic Equations

Cross-multiplication is a fundamental algebraic technique used to solve equations involving fractions. It simplifies the process by removing the denominators, thus converting the equation into a simpler one without fractions.
When two fractions are set equal to each other, as in \(\frac{(C+x)}{x-3}=\frac{x+8}{3}\), you can "cross-multiply" to eliminate the fractions.
Here’s how it works:

• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.

Then, you set the two products equal to each other:\[(C+x) \cdot 3 = (x+8) \cdot (x-3)\]
This removes the fraction entirely and results in a linear or polynomial equation that is easier to handle. Cross-multiplication helps streamline your calculations and is a first crucial step in solving equations like this one.

Expression Simplification

Expression simplification involves reducing complex algebraic expressions into their simplest form.
After cross-multiplying, the next challenge is to simplify the resulting equation. Consider the expression\[3C + 3x = x^2 - 3x + 8x - 24\].
You can simplify this by:

• Combining like terms: identify and add together terms that have the same variable raised to the same power.
• The terms \(-3x\) and \(8x\) combine to make \(5x\).

Now, the equation becomes:\[3C + 3x = x^2 + 5x - 24\]
Further simplification can involve isolating variables to solve for unknowns, which solidifies your understanding of how expressions can be manipulated for clarity and simplicity.

Polynomial Expansion and Its Role

Polynomial expansion is a technique used to distribute or expand products of sums into expressions with multiple terms. This technique is especially useful in cross-multiplication and is crucial in manipulating equations.
For example, the equation \((x-3)(x+8)\) needs to be expanded:

• Use distributive property (i.e., FOIL method): Multiply each term in the first binomial by each term in the second binomial.
• Multiply \(x \cdot x\), \(x \cdot 8\), \(-3 \cdot x\), and \(-3 \cdot 8\).

This results in:\[x^2 - 3x + 8x - 24\]
By applying polynomial expansion, we transform a compact expression into a form where all operations are explicit. This transformation is crucial for further simplification or equation solving. Understanding polynomial expansion is vital for students tackling algebraic problems involving multiple variables and expressions.