MTH 100 Chapter 5 & 6 Test Review – Step-by-Step Algebra Guidance

Q1. Factor completely: $8x^2y^3 + 10xy - 18xy$

Background

Topic: Factoring Polynomials

This question tests your ability to factor polynomials by first identifying the greatest common factor (GCF) and then factoring further if possible.

Key Terms and Formulas:

• Greatest Common Factor (GCF): The largest expression that divides all terms in the polynomial.

• Factoring: Writing a polynomial as a product of simpler polynomials.

Step-by-Step Guidance

1. Identify the GCF of all terms. Look for common factors in the coefficients and variables.

2. Factor out the GCF from each term.

3. After factoring out the GCF, examine the remaining polynomial to see if it can be factored further (e.g., as a trinomial or by grouping).

Try solving on your own before revealing the answer!

Q2. Factor completely: $100z^3 - 121$

Background

Topic: Factoring the Difference of Cubes or Squares

This question tests your ability to recognize and factor the difference of cubes or squares.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Check if both terms are perfect squares or perfect cubes.

2. If they are, rewrite each term as a square or cube.

3. Apply the appropriate factoring formula (difference of squares or cubes).

Try solving on your own before revealing the answer!

Q3. Factor completely: $6x^2 + 16x - 16$

Background

Topic: Factoring Trinomials

This question tests your ability to factor trinomials, especially when the leading coefficient is not 1.

Key Terms and Formulas:

• Trinomial Factoring: For $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$.

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the trinomial.

2. Multiply $a$ and $c$ to find the product.

3. Find two numbers that multiply to $ac$ and add to $b$.

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

Try solving on your own before revealing the answer!

Q4. Factor completely: $4ab - 15bc + c$

Background

Topic: Factoring by Grouping

This question tests your ability to factor polynomials with three terms, possibly by grouping or factoring out common factors.

Key Terms and Formulas:

• Factoring by Grouping: Group terms to factor common factors from each group.

Step-by-Step Guidance

1. Look for a common factor in all terms. If none, try grouping terms in pairs.

2. Factor out the GCF from each group.

3. Check if the grouped terms have a common binomial factor.

Try solving on your own before revealing the answer!

Q5. Factor completely: $2x^3 - 8x$

Background

Topic: Factoring Out the GCF and Difference of Squares

This question tests your ability to factor out the greatest common factor and then factor further if possible.

Key Terms and Formulas:

• GCF: The largest factor common to all terms.

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

Step-by-Step Guidance

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

2. Check if the remaining expression is a difference of squares or can be factored further.

Try solving on your own before revealing the answer!

Q6. Factor completely: $7x^2 - 21x - 70$

Background

Topic: Factoring Trinomials with a Common Factor

This question tests your ability to factor out a GCF and then factor a trinomial.

Key Terms and Formulas:

• GCF: The largest factor common to all terms.

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

Step-by-Step Guidance

1. Factor out the GCF from all terms.

2. Factor the resulting trinomial by finding two numbers that multiply to $ac$ and add to $b$.

3. Write the final factored form as a product of the GCF and the factored trinomial.

Try solving on your own before revealing the answer!

Q7. Factor completely: $4x^2 + 3x - 10$

Background

Topic: Factoring Trinomials

This question tests your ability to factor trinomials where the leading coefficient is not 1.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the trinomial.

2. Multiply $a$ and $c$ to find the product.

3. Find two numbers that multiply to $ac$ and add to $b$.

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

Try solving on your own before revealing the answer!

Q8. Factor completely: $64x^3 - 125$

Background

Topic: Factoring the Difference of Cubes

This question tests your ability to recognize and factor the difference of cubes.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Express each term as a cube (e.g., $64x^3 = (4x)^3$, $125 = 5^3$).

2. Apply the difference of cubes formula.

3. Simplify the resulting factors.

Try solving on your own before revealing the answer!

Q9. Factor completely: $10x^2 + 55x + 70$

Background

Topic: Factoring Trinomials with a Common Factor

This question tests your ability to factor out a GCF and then factor a trinomial.

Key Terms and Formulas:

• GCF: The largest factor common to all terms.

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

Step-by-Step Guidance

1. Factor out the GCF from all terms.

2. Factor the resulting trinomial by finding two numbers that multiply to $ac$ and add to $b$.

3. Write the final factored form as a product of the GCF and the factored trinomial.

Try solving on your own before revealing the answer!

Q10. For $\frac{11x^2 + 6x + 24}{x^2 + 2x + 4}$, find the numerical value when a) $x = 2$ and b) $x = -3$.

Background

Topic: Evaluating Rational Expressions

This question tests your ability to substitute values into a rational expression and simplify.

Key Terms and Formulas:

• Rational Expression: A fraction where the numerator and/or denominator are polynomials.

Step-by-Step Guidance

1. Substitute $x = 2$ into both the numerator and denominator, and simplify each part.

2. Write the simplified numerator and denominator as a fraction.

3. Repeat the process for $x = -3$.

Try solving on your own before revealing the answer!

Q11. For $\frac{x^2 - 4}{x}$, find any values for which the expression is undefined.

Background

Topic: Domain of Rational Expressions

This question tests your ability to determine when a rational expression is undefined (i.e., when the denominator is zero).

Key Terms and Formulas:

• Undefined Expression: Occurs when the denominator equals zero.

Step-by-Step Guidance

1. Set the denominator equal to zero: $x = 0$.

2. Solve for $x$ to find the value(s) that make the expression undefined.

Try solving on your own before revealing the answer!

Q12. For $\frac{x^2 + 3x + 4}{5x + 4}$, find any values for which the expression is undefined.

Background

Topic: Domain of Rational Expressions

This question tests your ability to determine when a rational expression is undefined.

Key Terms and Formulas:

• Undefined Expression: Occurs when the denominator equals zero.

Step-by-Step Guidance

1. Set the denominator equal to zero: $5x + 4 = 0$.

2. Solve for $x$ to find the value(s) that make the expression undefined.

Try solving on your own before revealing the answer!

Q13. Write the rational expression $\frac{75m^2 - 3}{25m - 5}$ in lowest terms.

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor numerators and denominators and reduce the expression to lowest terms.

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

• Reducing: Canceling common factors in the numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator and denominator completely.

2. Identify and cancel any common factors.

Try solving on your own before revealing the answer!

Q14. Write the rational expression $\frac{x^3 + 4x^2 + 12x}{x^2 + 4x + 4}$ in lowest terms.

Background

Topic: Simplifying Rational Expressions

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

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

• Reducing: Canceling common factors in the numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator and denominator completely.

2. Identify and cancel any common factors.

Try solving on your own before revealing the answer!

Q15. Write the rational expression $\frac{v^2 - 4}{v - 2}$ in lowest terms.

Background

Topic: Simplifying Rational Expressions

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

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Factor the numerator as a difference of squares.

2. Cancel any common factors with the denominator.

Try solving on your own before revealing the answer!

Q16. Simplify the complex fraction: $\frac{m^2 - 1}{8m} \div \frac{7}{64}$

Background

Topic: Simplifying Complex Fractions

This question tests your ability to divide rational expressions and simplify the result.

Key Terms and Formulas:

• Complex Fraction: A fraction where the numerator, denominator, or both are also fractions.

• Division of Fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Step-by-Step Guidance

1. Rewrite the division as multiplication by the reciprocal.

2. Factor the numerator and denominator where possible.

3. Simplify by canceling common factors.

Try solving on your own before revealing the answer!

Q17. Simplify the complex fraction: $\frac{a^2 - 1}{a + 1} \div \frac{a - 1}{g}$

Background

Topic: Simplifying Complex Fractions

This question tests your ability to divide rational expressions and simplify the result.

Key Terms and Formulas:

• Complex Fraction: A fraction where the numerator, denominator, or both are also fractions.

• Division of Fractions: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

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

Step-by-Step Guidance

1. Rewrite the division as multiplication by the reciprocal.

2. Factor the numerator if possible.

3. Simplify by canceling common factors.

Try solving on your own before revealing the answer!

Q18. Simplify the complex fraction: $\frac{6x^3 - 9x^2 + 3x}{9x^2 + 3x - 3}$

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor and reduce complex rational expressions.

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

• Reducing: Canceling common factors in the numerator and denominator.

Step-by-Step Guidance

1. Factor the numerator and denominator completely.

2. Identify and cancel any common factors.

Try solving on your own before revealing the answer!

Q19. Solve the equation: $16x^2 - 9 = 0$

Background

Topic: Solving Quadratic Equations by Factoring

This question tests your ability to solve quadratic equations by factoring and using the zero product property.

Key Terms and Formulas:

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

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

Step-by-Step Guidance

1. Rewrite the equation so that one side equals zero.

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!

Q20. Solve the equation: $6x^3 - 15x^2 - 9x = 0$

Background

Topic: Solving Polynomial Equations by Factoring

This question tests your ability to factor polynomials and solve for all possible values of $x$.

Key Terms and Formulas:

• Factoring: Expressing a polynomial as a product of its factors.

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

Step-by-Step Guidance

1. Set the equation equal to zero if it is not already.

2. Factor out the GCF from all terms.

3. Factor the remaining polynomial if possible.

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

Try solving on your own before revealing the answer!

Q21. Solve the equation: $7x^2 - 2x - 17 = 0$

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve quadratic equations, possibly by factoring or using the quadratic formula.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify $a$, $b$, and $c$ in the equation.

2. Attempt to factor the quadratic, or use the quadratic formula if factoring is difficult.

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

Try solving on your own before revealing the answer!

Q22. Solve the equation: $4x + 5 = -x$

Background

Topic: Solving Linear Equations

This question tests your ability to solve linear equations for $x$.

Key Terms and Formulas:

• Linear Equation: An equation of the form $ax + b = c$.

Step-by-Step Guidance

1. Combine like terms and isolate $x$ on one side of the equation.

2. Solve for $x$ by performing the necessary arithmetic operations.

Try solving on your own before revealing the answer!

Q23. Solve the equation: $\frac{y^2 - 12y + 144}{y^2 - 6y - 36} = 0$

Background

Topic: Solving Rational Equations

This question tests your ability to solve rational equations by setting the numerator equal to zero and checking for extraneous solutions.

Key Terms and Formulas:

• Rational Equation: An equation involving rational expressions.

• Zero of a Rational Expression: Occurs when the numerator is zero, provided the denominator is not zero at that value.

Step-by-Step Guidance

1. Set the numerator equal to zero and solve for $y$.

2. Check that the solution does not make the denominator zero.

Try solving on your own before revealing the answer!

Q24. In 2011, Trevor Bayne drove his Ford to victory in the Daytona 500 (mile) race. His time was 3.836 hours. What was his rate (to the nearest thousandth)?

Background

Topic: Distance, Rate, and Time

This question tests your ability to use the formula $d = rt$ to solve for rate.

Key Terms and Formulas:

• Distance Formula: $d = rt$

• Solving for Rate: $r = \frac{d}{t}$

Step-by-Step Guidance

1. Identify the total distance ($d = 500$ miles) and the time ($t = 3.836$ hours).

2. Set up the formula to solve for rate: $r = \frac{d}{t}$.

3. Plug in the values for $d$ and $t$.

Try solving on your own before revealing the answer!