College Algebra Review: Factoring, Quadratics, Rational Functions, and More

Q1. Factor completely: (a) $3x^3 - 2x^2 - 6x + 4$

Background

Topic: Polynomial Factoring

This question tests your ability to factor a cubic polynomial completely, possibly by grouping or other factoring techniques.

Key Terms and Formulas:

• Factoring by grouping: Rearranging and grouping terms to factor common factors.

• Factor theorem: If $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$.

Step-by-Step Guidance

1. Group the terms in pairs: $(3x^3 - 2x^2) + (-6x + 4)$.

2. Factor out the greatest common factor (GCF) from each group.

3. Look for a common binomial factor between the two groups.

4. Once factored, check if any quadratic or cubic factors can be further factored.

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Q2. Solve the equation by factoring: (a) $10x^3 - 40x = 0$

Background

Topic: Solving Polynomial Equations by Factoring

This question tests your ability to solve a cubic equation by factoring and using the zero product property.

Key Terms and Formulas:

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Factoring out the GCF.

Step-by-Step Guidance

1. Factor out the greatest common factor from the equation $10x^3 - 40x = 0$.

2. Set each factor equal to zero using the zero product property.

3. Solve each resulting equation for $x$.

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Q3. Find the domain of the rational function: (a) $f(x) = \frac{x-3}{x+2}$

Background

Topic: Domain of Rational Functions

This question tests your understanding of how to find the domain of a function that involves division by a variable expression.

Key Terms and Formulas:

• Domain: The set of all real numbers for which the function is defined.

• Rational function: A function of the form $\frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

Step-by-Step Guidance

1. Identify the denominator of the function: $x + 2$.

2. Set the denominator not equal to zero: $x + 2 \neq 0$.

3. Solve for $x$ to find the value(s) to exclude from the domain.

4. Express the domain in interval notation, excluding the value(s) found.

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Q4. Simplify the rational expression completely: (a) $\frac{x+1}{x^2-2x-3}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials and reduce rational expressions to lowest terms.

Key Terms and Formulas:

• Factoring quadratics: $x^2 - 2x - 3$ can be factored into two binomials.

• Reducing rational expressions: Cancel common factors in numerator and denominator.

Step-by-Step Guidance

1. Factor the denominator $x^2 - 2x - 3$.

2. Check if the numerator $x + 1$ shares any common factors with the denominator.

3. Cancel any common factors to simplify the expression.

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Q5. Find the least common denominator (LCD): (a) $\frac{4x}{x-2}$ and $\frac{x+1}{x-5}$

Background

Topic: Least Common Denominator of Rational Expressions

This question tests your ability to find the LCD for adding or subtracting rational expressions with different denominators.

Key Terms and Formulas:

• LCD: The smallest expression that is a multiple of each denominator.

Step-by-Step Guidance

1. Identify the denominators: $x-2$ and $x-5$.

2. Since the denominators are distinct and have no common factors, the LCD is their product.

3. Write the LCD as a product of the two denominators.

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Q6. Solve the rational equation and check for extraneous solutions: (a) $\frac{6}{x+3} = \frac{4}{x-3}$

Background

Topic: Solving Rational Equations

This question tests your ability to solve equations involving rational expressions and to check for extraneous solutions.

Key Terms and Formulas:

• Extraneous solution: A solution that does not satisfy the original equation due to restrictions (like division by zero).

• Cross-multiplication: Used to solve equations of the form $\frac{a}{b} = \frac{c}{d}$.

Step-by-Step Guidance

1. Cross-multiply to eliminate denominators: $6(x-3) = 4(x+3)$.

2. Expand both sides and collect like terms.

3. Solve for $x$.

4. Check your solution(s) in the original equation to ensure they do not make any denominator zero.

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Q7. A company is planning to manufacture small canoes. Fixed monthly cost is $20,000 and it costs $20 to produce each canoe.

Background

Topic: Cost Functions and Average Cost

This question tests your ability to write and analyze cost functions, including average cost, and solve for a specific average cost.

Key Terms and Formulas:

• Cost function: $C(x) = \text{fixed cost} + (\text{variable cost per unit}) \times x$

• Average cost function: $\overline{C}(x) = \frac{C(x)}{x}$

Step-by-Step Guidance

1. Write the total cost function $C(x)$ using the given fixed and variable costs.

2. Write the average cost function $\overline{C}(x)$ by dividing $C(x)$ by $x$.

3. Set $\overline{C}(x) = 40$ and solve for $x$ to find the number of canoes needed for an average cost of $40$ per canoe.

4. Be sure to include units in your answer.

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Q8. Evaluate each expression or state that the expression is not a real number: (a) $\sqrt{16}$, (b) $\sqrt{-16}$, (c) $-\sqrt{16}$, (d) $\sqrt{\frac{36}{49}}$, (e) $\sqrt{25} - \sqrt{16}$, (f) $\sqrt{25 - 16}$

Background

Topic: Evaluating Square Roots and Real vs. Non-Real Numbers

This question tests your understanding of square roots, including when the result is real or non-real (complex).

Key Terms and Formulas:

• Square root: $\sqrt{a}$ is the non-negative number whose square is $a$.

• For negative radicands, the result is not a real number (unless using complex numbers).

Step-by-Step Guidance

1. For each part, determine if the radicand (the number under the square root) is positive, zero, or negative.

2. Evaluate the square root if the radicand is non-negative.

3. If the radicand is negative, state that the expression is not a real number.

4. For expressions involving subtraction, perform the operations inside the radical first if needed.

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Q9. Let $f(x) = \sqrt{x-3}$. (a) Algebraically find the domain of $f(x)$. Report your answer in interval notation.

Background

Topic: Domain of Radical Functions

This question tests your ability to find the domain of a function involving a square root.

Key Terms and Formulas:

• Domain: The set of all real $x$ for which the function is defined.

• For $\sqrt{A}$, $A \geq 0$ for real values.

Step-by-Step Guidance

1. Set the radicand $x-3 \geq 0$.

2. Solve the inequality for $x$.

3. Express the solution in interval notation.

Try solving on your own before revealing the answer!