College Algebra Test #2 Review – Step-by-Step Study Guidance

Q1. Factor completely: $27a^4 - 12b^2$

Background

Topic: Factoring Polynomials

This question tests your ability to factor expressions by identifying common factors and using factoring techniques such as factoring out the greatest common factor (GCF) and recognizing special products.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest expression that divides each term.

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$

Step-by-Step Guidance

1. Identify the GCF of $27a^4$ and $12b^2$ and factor it out.

2. Examine the remaining expression to see if it can be factored further, possibly as a difference of squares or another special product.

3. Write the expression as a product of its factors, stopping before the final multiplication.

Try solving on your own before revealing the answer!

Final Answer:

$27a^4 - 12b^2 = 3(9a^4 - 4b^2) = 3(3a^2 - 2b)(3a^2 + 2b)$

We factored out the GCF (3), then recognized the difference of squares in the remaining expression.

Q2. Factor completely: $27a^3 - 8b^3$

Background

Topic: Factoring Sums and Differences of Cubes

This question tests your ability to factor expressions that are differences of cubes using the appropriate formula.

Key Terms and Formulas:

• Difference of Cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Step-by-Step Guidance

1. Recognize that $27a^3$ is $(3a)^3$ and $8b^3$ is $(2b)^3$.

2. Apply the difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, substituting $a = 3a$ and $b = 2b$.

3. Write out the factors, but do not expand or simplify the quadratic factor yet.

Try solving on your own before revealing the answer!

Final Answer:

$27a^3 - 8b^3 = (3a - 2b)(9a^2 + 6ab + 4b^2)$

We used the difference of cubes formula with $a = 3a$ and $b = 2b$.

Q3. Factor completely: $8x^2 + 18x + 9$

Background

Topic: Factoring Trinomials

This question tests your ability to factor a quadratic trinomial, possibly by grouping or by using the AC method.

Key Terms and Formulas:

• Quadratic Trinomial: $ax^2 + bx + c$

Step-by-Step Guidance

1. Identify $a = 8$, $b = 18$, $c = 9$.

2. Look for two numbers that multiply to $a \times c = 72$ and add to $b = 18$.

3. Rewrite the middle term using these two numbers and factor by grouping.

Try solving on your own before revealing the answer!

Final Answer:

$8x^2 + 18x + 9 = (4x + 3)(2x + 3)$

We split the middle term and factored by grouping.

Q4. Factor completely: $4x^3 + 4x^2 - 24x$

Background

Topic: Factoring Polynomials by GCF and Grouping

This question tests your ability to factor out the greatest common factor and then factor the remaining polynomial.

Key Terms and Formulas:

• GCF: The largest factor common to all terms.

Step-by-Step Guidance

1. Identify and factor out the GCF from all terms.

2. Factor the remaining quadratic expression, if possible.

3. Write the expression as a product of its factors, stopping before the final multiplication.

Try solving on your own before revealing the answer!

Final Answer:

$4x^3 + 4x^2 - 24x = 4x(x^2 + x - 6) = 4x(x + 3)(x - 2)$

We factored out $4x$ and then factored the quadratic.

Q5. Factor completely: $x^3 + 4x^2 - 4x - 16$

Background

Topic: Factoring by Grouping

This question tests your ability to factor a cubic polynomial by grouping terms and factoring common factors.

Key Terms and Formulas:

• Factoring by Grouping: Group terms to factor common factors, then factor further if possible.

Step-by-Step Guidance

1. Group the terms: $(x^3 + 4x^2) + (-4x - 16)$.

2. Factor out the common factor from each group.

3. Look for a common binomial factor and factor it out.

Try solving on your own before revealing the answer!

Final Answer:

$x^3 + 4x^2 - 4x - 16 = (x^2 - 4)(x + 4) = (x - 2)(x + 2)(x + 4)$

We grouped, factored, and then factored the difference of squares.

Q6. For the piecewise function $f(x) = \begin{cases} 9x + 1, & x < 7 \\ 7x, & 7 \leq x \leq 12 \\ 7 - 2x, & x > 12 \end{cases}$, find $2f(-7) + 3f(10)$

Background

Topic: Piecewise Functions and Function Evaluation

This question tests your ability to evaluate a piecewise function at specific values and combine the results.

Key Terms and Formulas:

• Piecewise Function: A function defined by different expressions for different intervals of the domain.

Step-by-Step Guidance

1. Determine which piece of the function to use for $f(-7)$ by checking which interval $-7$ falls into.

2. Evaluate $f(-7)$ using the appropriate formula.

3. Determine which piece to use for $f(10)$ and evaluate it.

4. Multiply $f(-7)$ by 2 and $f(10)$ by 3, then add the results.

Try solving on your own before revealing the answer!

Final Answer:

$f(-7) = 9(-7) + 1 = -62$ $f(10) = 7 \times 10 = 70$ $2f(-7) + 3f(10) = 2(-62) + 3(70) = -124 + 210 = 86$

We evaluated each function value and combined them as instructed.

Q7. Graph the function: $f(x) = \begin{cases} 3 - x, & x \leq 2 \\ 1 + 2x, & x > 2 \end{cases}$

Background

Topic: Graphing Piecewise Functions

This question tests your ability to graph a function defined by different expressions over different intervals.

Key Terms and Formulas:

• Piecewise Function: Different rules for different parts of the domain.

Step-by-Step Guidance

1. Identify the two pieces: $3 - x$ for $x \leq 2$ and $1 + 2x$ for $x > 2$.

2. Graph $y = 3 - x$ for $x \leq 2$. Find at least two points, including $x = 2$.

3. Graph $y = 1 + 2x$ for $x > 2$. Find at least two points, starting at $x = 2$ (open or closed dot as appropriate).

4. Check for continuity at $x = 2$ by evaluating both pieces at $x = 2$.

Try sketching the graph before revealing the answer!

Final Answer:

For $x \leq 2$, the graph is a line with slope $-1$ passing through $(2,1)$. For $x > 2$, the graph is a line with slope $2$starting just above$(2,5)$. The function is not continuous at$x = 2$.

Q8. Express the number in terms of $i$: $\sqrt{-112}$

Background

Topic: Complex Numbers and Imaginary Unit

This question tests your ability to express the square root of a negative number in terms of $i$.

Key Terms and Formulas:

• Imaginary Unit: $i = \sqrt{-1}$

• $\sqrt{-a} = i\sqrt{a}$ for $a > 0$

Step-by-Step Guidance

1. Rewrite $\sqrt{-112}$ as $\sqrt{-1 \times 112}$.

2. Express as $i\sqrt{112}$.

3. Simplify $\sqrt{112}$ by factoring out perfect squares.

Try simplifying before revealing the answer!

Final Answer:

$\sqrt{-112} = 4i\sqrt{7}$

We factored $112 = 16 \times 7$ and took the square root of $16$.

Q9. Express the number in terms of $i$: $-\sqrt{-49}$

Background

Topic: Complex Numbers and Imaginary Unit

This question tests your ability to express the square root of a negative number in terms of $i$ and handle the negative sign.

Key Terms and Formulas:

• $\sqrt{-a} = i\sqrt{a}$

Step-by-Step Guidance

1. Rewrite $-\sqrt{-49}$ as $-i\sqrt{49}$.

2. Simplify $\sqrt{49}$.

Try simplifying before revealing the answer!

Final Answer:

$-\sqrt{-49} = -7i$

We used $\sqrt{49} = 7$ and the definition of $i$.

Q10. Simplify and write in the form $a + bi$: $(-8 + 16i) - (6 - 3i)$

Background

Topic: Complex Number Arithmetic

This question tests your ability to subtract complex numbers and write the result in standard form.

Key Terms and Formulas:

• Complex Number: $a + bi$

• Subtract real and imaginary parts separately.

Step-by-Step Guidance

1. Distribute the negative sign to both terms in the second complex number.

2. Combine the real parts and the imaginary parts separately.

Try simplifying before revealing the answer!

Final Answer:

$(-8 + 16i) - (6 - 3i) = -14 + 19i$

We subtracted real and imaginary parts separately.

Q11. Simplify and write in the form $a + bi$: $(8 + 5i)^2$

Background

Topic: Complex Number Multiplication

This question tests your ability to square a complex number and simplify the result.

Key Terms and Formulas:

• $(a + b)^2 = a^2 + 2ab + b^2$

• $i^2 = -1$

Step-by-Step Guidance

1. Expand $(8 + 5i)^2$ using the binomial square formula.

2. Replace $i^2$ with $-1$ and combine like terms.

Try expanding before revealing the answer!

Final Answer:

$(8 + 5i)^2 = 39 + 80i$

We expanded and simplified using $i^2 = -1$.

Q12. Simplify and write in the form $a + bi$: $(9 - 7i)(4 + 9i)$

Background

Topic: Multiplying Complex Numbers

This question tests your ability to multiply two complex numbers and simplify the result.

Key Terms and Formulas:

• FOIL method for binomials

• $i^2 = -1$

Step-by-Step Guidance

1. Multiply each term in the first binomial by each term in the second binomial (FOIL).

2. Combine like terms and replace $i^2$ with $-1$.

Try multiplying before revealing the answer!

Final Answer:

$(9 - 7i)(4 + 9i) = 99 + 59i$

We used the distributive property and simplified.

Q13. Simplify and write in the form $a + bi$: $\frac{9 - i}{-4 + 6i}$

Background

Topic: Dividing Complex Numbers

This question tests your ability to divide complex numbers by multiplying numerator and denominator by the conjugate of the denominator.

Key Terms and Formulas:

• Conjugate: For $a + bi$, the conjugate is $a - bi$.

• Multiply numerator and denominator by the conjugate of the denominator.

Step-by-Step Guidance

1. Find the conjugate of the denominator $-4 + 6i$.

2. Multiply both numerator and denominator by this conjugate.

3. Expand both numerator and denominator, and simplify using $i^2 = -1$.

4. Write the result in the form $a + bi$.

Try working through the multiplication before revealing the answer!

Final Answer:

$\frac{9 - i}{-4 + 6i} = -1 + \frac{5}{13}i$

We multiplied by the conjugate and simplified.

Q14. Simplify: $i^{463}$

Background

Topic: Powers of $i$

This question tests your ability to simplify powers of the imaginary unit $i$ using its cyclical pattern.

Key Terms and Formulas:

• $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$ (cycle repeats every 4)

Step-by-Step Guidance

1. Divide the exponent by 4 and find the remainder.

2. Use the remainder to determine the equivalent power of $i$.

Try finding the remainder before revealing the answer!

Final Answer:

$i^{463} = i^{3} = -i$

The remainder is 3, so the answer is $-i$.

Q15. Solve: $x^2 + 8x + 16 = 10$

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve a quadratic equation by moving all terms to one side and factoring or using the quadratic formula.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Subtract 10 from both sides to set the equation to zero.

2. Factor the resulting quadratic, if possible.

3. If factoring is not possible, use the quadratic formula to solve for $x$.

Try factoring or setting up the quadratic formula before revealing the answer!

Final Answer:

$x^2 + 8x + 16 = 10 \implies x^2 + 8x + 6 = 0$ Using the quadratic formula: $x = \frac{-8 \pm \sqrt{64 - 24}}{2} = \frac{-8 \pm \sqrt{40}}{2} = -4 \pm \sqrt{10}$

Q16. Solve: $x^2 + 4x + 68 = 0$

Background

Topic: Solving Quadratic Equations with Complex Solutions

This question tests your ability to solve a quadratic equation that has complex (non-real) solutions.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify $a = 1$, $b = 4$, $c = 68$.

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

3. Since the discriminant is negative, express the solution in terms of $i$.

Try setting up the quadratic formula before revealing the answer!

Final Answer:

$x = \frac{-4 \pm \sqrt{16 - 272}}{2} = \frac{-4 \pm \sqrt{-256}}{2} = \frac{-4 \pm 16i}{2} = -2 \pm 8i$

Q17. Use the quadratic formula to find the exact solutions: $3x^2 - 14x - 5 = 0$

Background

Topic: Quadratic Formula

This question tests your ability to apply the quadratic formula to find the exact solutions of a quadratic equation.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify $a = 3$, $b = -14$, $c = -5$.

2. Plug these values into the quadratic formula.

3. Calculate the discriminant $(-14)^2 - 4 \cdot 3 \cdot (-5)$.

4. Set up the expression for $x$ using the calculated discriminant, but do not simplify further.

Try plugging in the values before revealing the answer!

Final Answer:

$x = \frac{14 \pm \sqrt{196 + 60}}{6} = \frac{14 \pm \sqrt{256}}{6} = \frac{14 \pm 16}{6}$ So, $x = 5$ or $x = -\frac{1}{3}$