College Algebra Exam 2 Review: Step-by-Step Guidance

Q1. Solve by factoring: $x^2 - 9x + 20 = 0$

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to factor quadratic expressions and use the zero-product property to solve for $x$.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring: Expressing the quadratic as a product of two binomials

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

Step-by-Step Guidance

1. Write the equation in standard form: $x^2 - 9x + 20 = 0$.

2. Identify two numbers that multiply to $20$and add to$-9$.

3. Rewrite the quadratic as $(x - a)(x - b) = 0$, where $a$ and $b$ are the numbers found in step 2.

4. Apply the zero-product property to set each factor equal to zero.

Try solving on your own before revealing the answer!

Q2. Solve by factoring: $3x^2 + 11x = 4$

Background

Topic: Solving Quadratic Equations by Factoring

This question asks you to rearrange the equation and factor to find the solutions for $x$.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring and Zero-Product Property

Step-by-Step Guidance

1. Move all terms to one side: $3x^2 + 11x - 4 = 0$.

2. Factor the quadratic expression.

3. Set each factor equal to zero and solve for $x$.

Try solving on your own before revealing the answer!

Q3. Solve by the Quadratic Formula: $10z^2 + 6z = 7$

Background

Topic: Quadratic Formula

This question tests your ability to use the quadratic formula to solve for $z$.

Key Terms and Formulas

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

• Standard Form: $az^2 + bz + c = 0$

Step-by-Step Guidance

1. Rewrite the equation in standard form: $10z^2 + 6z - 7 = 0$.

2. Identify $a = 10$, $b = 6$, $c = -7$.

3. Plug these values into the quadratic formula.

4. Calculate the discriminant: $b^2 - 4ac$.

Try solving on your own before revealing the answer!

Q4. Solve by Quadratic Formula: $2x^2 + 20x + 32 = 0$

Background

Topic: Quadratic Formula

This question requires you to use the quadratic formula to solve for $x$.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Identify $a = 2$, $b = 20$, $c = 32$.

2. Plug these values into the quadratic formula.

3. Calculate the discriminant: $b^2 - 4ac$.

Try solving on your own before revealing the answer!

Q5. Solve the quadratic equation: $2x^2 = 4x - 3$

Background

Topic: Solving Quadratic Equations

This question tests your ability to rearrange and solve a quadratic equation.

Key Terms and Formulas

• Quadratic Equation: $ax^2 + bx + c = 0$

• Factoring or Quadratic Formula

Step-by-Step Guidance

1. Move all terms to one side: $2x^2 - 4x + 3 = 0$.

2. Identify $a$, $b$, and $c$.

3. Decide whether to factor or use the quadratic formula.

Try solving on your own before revealing the answer!

Q6. Given the function $f(x) = -18 + 0.5x - \pi x^2$

Background

Topic: Polynomial Functions

This question asks you to analyze the structure and properties of a polynomial function.

Key Terms and Formulas

• Leading Term: The term with the highest degree

• Leading Coefficient: The coefficient of the leading term

• Degree: The highest power of $x$

• Classification: Constant, quadratic, cubic, quartic

• End Behavior: How the function behaves as $x \to \pm\infty$

Step-by-Step Guidance

1. Identify the term with the highest power of $x$ (leading term).

2. State the coefficient of the leading term (leading coefficient).

3. Determine the degree of the polynomial.

4. Classify the polynomial based on its degree.

5. Analyze the sign of the leading coefficient to determine end behavior.

Try answering each part before revealing the answer!

Q7. Given the function $f(x) = -2x^2 - 16x - 31$

Background

Topic: Quadratic Functions and Their Graphs

This question asks you to find key features of a quadratic function, including vertex, axis of symmetry, y-intercept, and intervals of increase/decrease.

Key Terms and Formulas

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

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Y-intercept: $f(0)$

• Maximum/Minimum: Depends on the sign of $a$

• Intervals of Increase/Decrease: Based on vertex and direction of parabola

Step-by-Step Guidance

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

2. Calculate the vertex by plugging the axis of symmetry into $f(x)$.

3. Find the y-intercept by evaluating $f(0)$.

4. Determine if the parabola opens up or down (sign of $a$).

5. Use the vertex to describe intervals of increase and decrease.

Try solving each part before revealing the answer!

Q8. Given the function $f(x) = -2x(x-7)x + 42$

Background

Topic: Polynomial Functions and Their Graphs

This question asks you to analyze zeros, multiplicity, degree, leading coefficient, and end behavior of a polynomial.

Key Terms and Formulas

• Zeros: Values of $x$ where $f(x) = 0$

• Multiplicity: Number of times a zero is repeated

• Degree: Highest power of $x$

• Leading Coefficient: Coefficient of the highest degree term

• End Behavior: Based on degree and leading coefficient

Step-by-Step Guidance

1. Expand the function to identify all terms.

2. Set $f(x) = 0$ and solve for zeros and their multiplicities.

3. Determine the degree by finding the highest power of $x$.

4. Identify the leading coefficient.

5. Analyze end behavior using degree and leading coefficient.

Try answering each part before revealing the answer!

Q9. Determine the maximum number of zeros, x-intercepts, and turning points for $f(x) = 2x^3 + 2x^2 - 9x + 4$

Background

Topic: Polynomial Function Properties

This question tests your understanding of the relationship between the degree of a polynomial and its zeros and turning points.

Key Terms and Formulas

• Degree: Highest power of $x$

• Maximum number of zeros: Equal to the degree

• Maximum number of turning points: Degree minus one

Step-by-Step Guidance

1. Identify the degree of the polynomial.

2. State the maximum possible number of zeros.

3. State the maximum possible number of turning points.

Try answering before revealing the answer!

Q10. Given the function $f(x) = x^4 - x^3 - 20x^2$

Background

Topic: Polynomial Classification and Factoring

This question asks you to name the polynomial and factor it completely to find zeros and their multiplicities.

Key Terms and Formulas

• Quartic Polynomial: Degree 4

• Factoring: Expressing as a product of polynomials

• Zeros and Multiplicity

Step-by-Step Guidance

1. Identify the degree and name the polynomial.

2. Factor out common terms.

3. Factor the remaining quadratic or cubic expression.

4. List the zeros and their multiplicities.

Try answering before revealing the answer!

Q11. Use Synthetic Division to determine whether the binomial is a factor of $f(x) = x^3 - 11x^2 + 17x + 90$; $x + 6$

Background

Topic: Synthetic Division and Factor Theorem

This question tests your ability to use synthetic division to check if a binomial is a factor of a polynomial.

Key Terms and Formulas

• Synthetic Division: Shortcut for dividing polynomials

• Factor Theorem: If $f(c) = 0$, then $x - c$ is a factor

Step-by-Step Guidance

1. Set $x + 6 = 0$ to find $x = -6$.

2. Use synthetic division with $-6$ and the coefficients of $f(x)$.

3. Check the remainder to determine if $x + 6$ is a factor.

Try performing synthetic division before revealing the answer!

Q12. Use synthetic division to find the function value: $f(x) = 2x^3 + 2x^2 + 4x + 8$; find $f(-3)$

Background

Topic: Synthetic Division for Function Evaluation

This question asks you to use synthetic division to evaluate a polynomial at a specific value.

Key Terms and Formulas

• Synthetic Division

• Function Value: $f(c)$

Step-by-Step Guidance

1. Set up synthetic division using $-3$ and the coefficients of $f(x)$.

2. Perform synthetic division to find the remainder, which is $f(-3)$.

Try performing synthetic division before revealing the answer!

Q13. Factor the polynomial completely and list the zeros: $f(x) = x^3 + 3x^2 - 10x - 24$

Background

Topic: Factoring Cubic Polynomials

This question tests your ability to factor a cubic polynomial and find its zeros.

Key Terms and Formulas

• Factoring by grouping or synthetic division

• Zeros: Values of $x$ where $f(x) = 0$

Step-by-Step Guidance

1. Try factoring by grouping or use synthetic division to find one zero.

2. Factor the remaining quadratic.

3. List all zeros.

Try factoring before revealing the answer!

Q14. Determine the domain of $f(x) = 3x^2 - 8x + 15$

Background

Topic: Domain of Polynomial Functions

This question asks you to determine the set of all possible input values for a quadratic function.

Key Terms and Formulas

• Domain: Set of all real numbers for polynomials

Step-by-Step Guidance

1. Recognize that quadratic functions are defined for all real numbers.

Try stating the domain before revealing the answer!

Q15. Given the rational function $f(x) = \frac{x^2 - 4}{x^2 + 6x + 8}$

Background

Topic: Rational Functions and Asymptotes

This question asks you to analyze vertical and horizontal asymptotes, holes, and intercepts of a rational function.

Key Terms and Formulas

• Vertical Asymptote: Set denominator equal to zero

• Hole: Common factor in numerator and denominator

• Horizontal Asymptote: Compare degrees of numerator and denominator

• Intercepts: Set $x = 0$ for y-intercept, numerator = 0 for x-intercept

Step-by-Step Guidance

1. Factor numerator and denominator.

2. Identify values that make the denominator zero (vertical asymptotes).

3. Check for common factors (holes).

4. Determine horizontal asymptote by comparing degrees.

5. Find x- and y-intercepts.

Try answering each part before revealing the answer!

Q16. Given the rational function $f(x) = \frac{3x}{x^2 - 4}$

Background

Topic: Rational Functions and Asymptotes

This question asks you to analyze vertical and horizontal asymptotes, holes, and intercepts of a rational function.

Key Terms and Formulas

• Vertical Asymptote: Set denominator equal to zero

• Hole: Common factor in numerator and denominator

• Horizontal Asymptote: Compare degrees of numerator and denominator

• Intercepts: Set $x = 0$ for y-intercept, numerator = 0 for x-intercept

Step-by-Step Guidance

1. Factor denominator.

2. Identify values that make the denominator zero (vertical asymptotes).

3. Check for common factors (holes).

4. Determine horizontal asymptote by comparing degrees.

5. Find x- and y-intercepts.

Try answering each part before revealing the answer!