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 $x - a$ is a factor, then $f(a) = 0$.

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 you have a product of two factors, check if either can be factored further.

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

Background

Topic: Solving Polynomial Equations by Factoring

This question asks you to solve a cubic equation by factoring and finding all real solutions.

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 each 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 rational function by identifying values that make the denominator zero.

Key Terms and Formulas:

• Rational function: A function of the form $f(x) = \frac{P(x)}{Q(x)}$.

• Domain: All real numbers except those that make the denominator zero.

Step-by-Step Guidance

1. Set the denominator $x + 2$ equal to zero and solve for $x$.

2. Exclude this value from the set of all real numbers.

3. Express the domain in interval notation.

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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: 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: (a) $\frac{4x}{x-2}$ and $\frac{x+1}{x-5}$

Background

Topic: Least Common Denominator (LCD) of Rational Expressions

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

Key Terms and Formulas:

• LCD: The least common multiple of the denominators.

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 single expression.

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Q6. Solve the rational equation. 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:

• Cross-multiplication: If $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$ (provided $b, d \neq 0$).

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

Step-by-Step Guidance

1. Cross-multiply to eliminate the denominators.

2. Solve the resulting linear equation for $x$.

3. Check your solution by substituting back into the original equation to ensure no division by zero occurs.

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

• (a) Write the cost function, $C(x)$, of producing $x$ canoes.

• (b) Write the average cost function, $\overline{C}(x)$, of producing $x$ canoes.

• (c) How many canoes must be produced each month for the company to have an average cost of $40 per canoe?

Background

Topic: Cost Functions and Average Cost

This question tests your understanding of linear cost functions and how to find average cost and solve for a specific value.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

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

2. Express the average cost function $\overline{C}(x)$ as $\frac{C(x)}{x}$.

3. Set $\overline{C}(x) = 40$ and solve for $x$.

4. Isolate $x$ and simplify the equation to find the required number of canoes.

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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 ability to evaluate square roots and recognize when an expression is not a real number.

Key Terms and Formulas:

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

• Imaginary unit: $i = \sqrt{-1}$, used when taking the square root of a negative number.

Step-by-Step Guidance

1. Evaluate each square root, noting if the radicand (the number under the root) is positive or negative.

2. For negative radicands, express the answer in terms of $i$ if not a real number.

3. For expressions involving subtraction, perform the operations in the correct order.

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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.

• (b) Evaluate $f(3)$, $f(4)$, $f(7)$, $f(12)$.

• (c) Use part (b) to sketch the graph of $f(x)$.

Background

Topic: Domain of Radical Functions and Function Evaluation

This question tests your understanding of domains for square root functions and evaluating function values.

Key Terms and Formulas:

• Domain: Set the radicand $x-3 \geq 0$ to find valid $x$ values.

• Function evaluation: Substitute the given $x$ values into $f(x)$.

Step-by-Step Guidance

1. Set $x-3 \geq 0$ and solve for $x$ to find the domain.

2. Substitute $x = 3, 4, 7, 12$ into $f(x)$ and simplify each result.

3. Use the points $(x, f(x))$ to help sketch the graph.

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Q10. Evaluate each cube root: (a) $\sqrt[3]{64}$, (b) $\sqrt[3]{-\frac{8}{27}}$

Background

Topic: Cube Roots

This question tests your ability to evaluate cube roots, including those of negative numbers and fractions.

Key Terms and Formulas:

• Cube root: $\sqrt[3]{a}$ is the number that, when cubed, gives $a$.

• Cube roots of negative numbers are real.

Step-by-Step Guidance

1. For $\sqrt[3]{64}$, find the number that when raised to the third power equals 64.

2. For $\sqrt[3]{-\frac{8}{27}}$, find the cube root of the numerator and denominator separately, keeping the negative sign.

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Q11. Use radical notation to rewrite each expression. Simplify, if possible: (a) $(100)^{1/2}$, (b) $(-64)^{1/3}$, (c) $(25)^{3/2}$

Background

Topic: Rational Exponents and Radical Notation

This question tests your ability to convert between rational exponents and radical notation, and to simplify.

Key Terms and Formulas:

• Rational exponent: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Step-by-Step Guidance

1. Rewrite each expression using radical notation based on the exponent.

2. Simplify the radical if possible.

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Q12. Rewrite each expression with rational exponents: (a) $\sqrt{11}$, (b) $\sqrt[5]{3x}$, (c) $\sqrt[3]{x^2}$, (d) $(\sqrt{7xy})^3$

Background

Topic: Radical and Rational Exponents

This question tests your ability to express radicals as exponents.

Key Terms and Formulas:

• $\sqrt[n]{a^m} = a^{m/n}$

Step-by-Step Guidance

1. For each radical, write the equivalent expression using rational exponents.

2. Apply exponent rules to simplify if possible.

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Q13. Rewrite each expression with a positive exponent. Simplify if possible: (a) $9^{-1/2}$, (b) $(-27)^{-2/3}$, (c) $\left(\frac{8}{125}\right)^{-1/3}$

Background

Topic: Negative and Rational Exponents

This question tests your understanding of negative exponents and how to rewrite them as positive exponents.

Key Terms and Formulas:

• $a^{-n} = \frac{1}{a^n}$

• $a^{m/n} = \sqrt[n]{a^m}$

Step-by-Step Guidance

1. Rewrite each expression with a positive exponent using the negative exponent rule.

2. Simplify the resulting expression if possible.

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Q14. Simplify the following. Write non-real answers using the form $a + bi$: (a) $\sqrt{28}$, (b) $\sqrt{75}$, (c) $\sqrt{-36}$, (d) $\sqrt{-63}$, (e) $7 + \sqrt{18}$, (f) $-3 - \sqrt{-500}$

Background

Topic: Simplifying Radicals and Complex Numbers

This question tests your ability to simplify square roots, including those with negative radicands, and to express non-real answers in standard form.

Key Terms and Formulas:

• For $\sqrt{a}$, factor $a$ to simplify.

• For $\sqrt{-a}$, write as $i\sqrt{a}$.

Step-by-Step Guidance

1. Factor each radicand to simplify the square root.

2. For negative radicands, express the answer in terms of $i$.

3. Combine like terms where possible.

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Q15. Solve each equation by the square root property. Simplify solutions if possible. Express imaginary solutions in the form $a + bi$: (a) $5x^2 + 9 = 29$, (b) $(x-5)^2 = -4$

Background

Topic: Solving Quadratic Equations by Square Root Property

This question tests your ability to isolate the squared term and apply the square root property, including handling imaginary solutions.

Key Terms and Formulas:

• Square root property: If $x^2 = a$, then $x = \pm\sqrt{a}$

Step-by-Step Guidance

1. Isolate the squared term on one side of the equation.

2. Apply the square root property to both sides.

3. Simplify the square root, expressing imaginary solutions as needed.

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Q16. A rectangular park is 6 miles long and 3 miles wide. How long is a pedestrian route that runs diagonally across the park? Round your answer to the nearest tenth. Report your answer with appropriate units.

Background

Topic: Pythagorean Theorem

This question tests your ability to apply the Pythagorean theorem to find the length of the diagonal of a rectangle.

Key Terms and Formulas:

• Pythagorean theorem: $c^2 = a^2 + b^2$

Step-by-Step Guidance

1. Let $a = 6$ miles and $b = 3$ miles.

2. Plug these values into the Pythagorean theorem to solve for $c$ (the diagonal).

3. Take the square root to find the length of the diagonal.

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Q17. Solve each equation using the quadratic formula. Simplify solutions if possible. Express imaginary solutions in the form $a + bi$: (a) $5x^2 + 8x = -3$, (b) $2x^2 = -4x + 5$, (c) $x^2 + 6x + 13 = 0$

Background

Topic: Quadratic Formula

This question tests your ability to use the quadratic formula to solve quadratic equations, including those with complex solutions.

Key Terms and Formulas:

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Step-by-Step Guidance

1. Rewrite each equation in standard form $ax^2 + bx + c = 0$.

2. Identify $a$, $b$, and $c$ for each equation.

3. Plug these values into the quadratic formula.

4. Simplify under the square root (the discriminant) to determine if solutions are real or complex.

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Q18. Consider the function $f(x) = (x + 3)^2 - 9$.

• (a) Algebraically find the vertex of $f(x)$.

• (b) Algebraically find the $x$-intercepts of $f(x)$.

• (c) Algebraically find the $y$-intercept of $f(x)$.

• (d) Sketch the graph of $f(x)$. Be sure to include all information from points (a) – (c).

• (e) Find the range of $f(x)$.

Background

Topic: Quadratic Functions – Vertex Form, Intercepts, and Range

This question tests your understanding of the vertex form of a quadratic, finding intercepts, and determining the range.

Key Terms and Formulas:

• Vertex form: $f(x) = a(x - h)^2 + k$; vertex at $(h, k)$

• $x$-intercepts: Set $f(x) = 0$ and solve for $x$

• $y$-intercept: Set $x = 0$ and solve for $f(0)$

Step-by-Step Guidance

1. Identify the vertex from the vertex form.

2. Set $f(x) = 0$ and solve for $x$ to find $x$-intercepts.

3. Set $x = 0$ to find the $y$-intercept.

4. Use this information to sketch the graph and determine the range.

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Q19. Consider the function $f(x) = 2x^2 - 12x + 10$.

• (a) Algebraically find the vertex of $f(x)$.

• (b) Algebraically find the $x$-intercepts of $f(x)$.

• (c) Algebraically find the $y$-intercept of $f(x)$.

• (d) Sketch the graph of $f(x)$. Be sure to include all information from points (a) – (c).

• (f) Find the range of $f(x)$.

Background

Topic: Quadratic Functions – Standard Form, Vertex, Intercepts, and Range

This question tests your ability to analyze a quadratic function in standard form.

Key Terms and Formulas:

• Vertex: $x = -\frac{b}{2a}$, $y = f(x)$

• $x$-intercepts: Set $f(x) = 0$ and solve for $x$

• $y$-intercept: Set $x = 0$ and solve for $f(0)$

Step-by-Step Guidance

1. Find the vertex using $x = -\frac{b}{2a}$.

2. Set $f(x) = 0$ and solve for $x$ to find $x$-intercepts.

3. Set $x = 0$ to find the $y$-intercept.

4. Use this information to sketch the graph and determine the range.

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Q20. Consider the quadratic function $f(x) = -3x^2 + 12x + 1$.

• (a) Without graphing, determine whether the function has a maximum value or a minimum value. Justify your answer.

• (b) Find the maximum or minimum value and determine where it occurs.

• (c) Identify the function’s domain and range.

Background

Topic: Quadratic Functions – Maximum/Minimum, Domain, and Range

This question tests your understanding of the direction of a parabola and how to find its vertex, domain, and range.

Key Terms and Formulas:

• If $a < 0$, the parabola opens downward (maximum); if $a > 0$, it opens upward (minimum).

• Vertex: $x = -\frac{b}{2a}$

• Domain: All real numbers for quadratics.

Step-by-Step Guidance

1. Identify the leading coefficient $a$ to determine if the function has a maximum or minimum.

2. Find the $x$-coordinate of the vertex using $x = -\frac{b}{2a}$.

3. Plug this $x$ value into $f(x)$ to find the maximum or minimum value.

4. State the domain and use the vertex to help determine the range.

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Q21. A person standing close to the edge of a 200-foot building throws a baseball vertically upward. The quadratic function $s(t) = -16t^2 + 64t + 200$ models the ball’s height above the ground, $s(t)$, in feet, $t$ seconds after it was thrown.

• (a) After how many seconds does the ball reach its maximum height? What is the maximum height? Report your answer with appropriate units.

• (b) How many seconds does it take until the ball finally hits the ground? Round your answer to the nearest tenth. Report your answer with appropriate units.

• (c) Find $s(0)$ and describe what this means.

Background

Topic: Quadratic Applications – Projectile Motion

This question tests your ability to analyze a quadratic function modeling vertical motion, including finding maximum height and interpreting initial conditions.

Key Terms and Formulas:

• Maximum height occurs at $t = -\frac{b}{2a}$ for $s(t) = at^2 + bt + c$

• To find when the ball hits the ground, set $s(t) = 0$ and solve for $t$

• $s(0)$ gives the initial height

Step-by-Step Guidance

1. Find the time at which the maximum height occurs using $t = -\frac{b}{2a}$.

2. Plug this $t$ value into $s(t)$ to find the maximum height.

3. Set $s(t) = 0$ and solve the quadratic equation for $t$ to find when the ball hits the ground.

4. Evaluate $s(0)$ and interpret its meaning in the context of the problem.

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