College Algebra Practice Test Guidance

Study Guide - Smart Notes

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Q1. Determine the domain of the variable x in the expression \( \frac{x}{x+57} \).

Background

Topic: Domain of Rational Expressions

This question tests your understanding of how to find the domain of a rational expression, which involves identifying values that make the denominator zero (undefined).

Key Terms and Formulas:

Domain: The set of all real numbers for which the expression is defined.
Rational Expression: An expression of the form \( \frac{P(x)}{Q(x)} \), where \( Q(x) \neq 0 \).

Step-by-Step Guidance

Identify the denominator of the expression: \( x + 57 \).
Set the denominator equal to zero to find values that are not allowed: \( x + 57 = 0 \).
Solve for \( x \) to determine which value makes the denominator zero.
Exclude this value from the set of all real numbers to write the domain.

Try solving on your own before revealing the answer!

Final Answer:

The domain is all real numbers except \( x = -57 \).

So, the domain is \( \{ x \mid x \neq -57 \} \).

Q2. Reduce the rational expression \( \frac{x^2 + 3x}{3x + 9} \) to lowest terms.

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials and cancel common factors in a rational expression.

Key Terms and Formulas:

Factor: Write a polynomial as a product of its factors.
Lowest Terms: A rational expression is in lowest terms when the numerator and denominator have no common factors (other than 1).

Step-by-Step Guidance

Factor the numerator: \( x^2 + 3x \).
Factor the denominator: \( 3x + 9 \).
Write the expression with the factored numerator and denominator.
Identify and cancel any common factors between numerator and denominator.

Try solving on your own before revealing the answer!

Final Answer:

\( \frac{x^2 + 3x}{3x + 9} = \frac{x(x+3)}{3(x+3)} = \frac{x}{3} \) for \( x \neq -3 \).

We factored both numerator and denominator, then cancelled the common factor \( x+3 \).

Q3. Simplify the expression \( \frac{x^7 y^3}{x^5 y^6} \) (Assume all bases are nonzero; write with positive exponents).

Background

Topic: Laws of Exponents

This question tests your ability to apply exponent rules to simplify expressions with variables in the numerator and denominator.

Key Terms and Formulas:

Quotient Rule for Exponents: \( \frac{a^m}{a^n} = a^{m-n} \)
Positive Exponents: Express your answer with only positive exponents.

Step-by-Step Guidance

Apply the quotient rule to the \( x \) terms: \( x^7 / x^5 \).
Apply the quotient rule to the \( y \) terms: \( y^3 / y^6 \).
Combine the results to write the simplified expression.
If any exponents are negative, rewrite them as positive exponents in the denominator.

Try solving on your own before revealing the answer!

Final Answer:

\( \frac{x^7 y^3}{x^5 y^6} = x^{2} y^{-3} = \frac{x^2}{y^3} \)

We subtracted exponents for each variable and rewrote negative exponents as positive in the denominator.

Q4. Reduce the rational expression \( \frac{4t - 12}{t^2 - 9} \) to lowest terms.

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials and simplify rational expressions by cancelling common factors.

Key Terms and Formulas:

Factoring: Expressing a polynomial as a product of its factors.
Difference of Squares: \( a^2 - b^2 = (a-b)(a+b) \)

Step-by-Step Guidance

Factor the numerator: \( 4t - 12 \).
Factor the denominator: \( t^2 - 9 \).
Write the expression with the factored numerator and denominator.
Identify and cancel any common factors.

Try solving on your own before revealing the answer!

Final Answer:

\( \frac{4t - 12}{t^2 - 9} = \frac{4(t-3)}{(t-3)(t+3)} = \frac{4}{t+3} \) for \( t \neq 3, -3 \).

We factored both numerator and denominator and cancelled the common factor \( t-3 \).

Q5. Factor the polynomial \( x^2 - 14x + 48 \).

Background

Topic: Factoring Quadratic Polynomials

This question tests your ability to factor a quadratic expression into two binomials.

Key Terms and Formulas:

Quadratic: An expression of the form \( ax^2 + bx + c \).
Factoring: Find two numbers that multiply to \( c \) and add to \( b \).

Step-by-Step Guidance

Identify \( a = 1 \), \( b = -14 \), \( c = 48 \).
Find two numbers whose product is 48 and whose sum is -14.
Write the factors as \( (x - m)(x - n) \), where \( m \) and \( n \) are the numbers found.

Try solving on your own before revealing the answer!

Final Answer:

\( x^2 - 14x + 48 = (x - 8)(x - 6) \)

Because -8 and -6 multiply to 48 and add to -14.

Q6. Graph the numbers \( x \) on the real number line for \( x \geq -2 \).

Background

Topic: Graphing Inequalities

This question tests your ability to represent solutions to inequalities on the real number line.

Key Terms and Formulas:

Inequality: A mathematical statement that relates expressions that are not necessarily equal.
Graphing: Use a closed circle for \( \geq \) or \( \leq \), and shade to the right for \( \geq \).

Step-by-Step Guidance

Locate -2 on the real number line.
Since the inequality is \( x \geq -2 \), use a closed circle at -2.
Shade the number line to the right of -2 to indicate all numbers greater than or equal to -2.
Compare the given graph choices to see which matches this description.

Try solving on your own before revealing the answer!

Final Answer:

The correct graph is the one with a closed circle at -2 and shading to the right (greater than or equal to -2).

Q7. Factor the trinomial \( x^2 - x - 30 \).

Background

Topic: Factoring Quadratic Trinomials

This question tests your ability to factor a quadratic trinomial into two binomials.

Key Terms and Formulas:

Quadratic Trinomial: \( ax^2 + bx + c \)
Factoring: Find two numbers that multiply to \( c \) and add to \( b \).

Step-by-Step Guidance

Identify \( a = 1 \), \( b = -1 \), \( c = -30 \).
Find two numbers whose product is -30 and whose sum is -1.
Write the factors as \( (x + m)(x + n) \), where \( m \) and \( n \) are the numbers found.

Try solving on your own before revealing the answer!

Final Answer:

\( x^2 - x - 30 = (x + 5)(x - 6) \)

Because 5 and -6 multiply to -30 and add to -1.

Q8. Simplify the expression \( 8^{-2} \) (Type an integer or a simplified fraction).

Background

Topic: Negative Exponents

This question tests your understanding of how to rewrite expressions with negative exponents as positive exponents (reciprocals).

Key Terms and Formulas:

Negative Exponent Rule: \( a^{-n} = \frac{1}{a^n} \)

Step-by-Step Guidance

Recognize that a negative exponent means take the reciprocal of the base raised to the positive exponent.
Rewrite \( 8^{-2} \) as \( \frac{1}{8^2} \).
Calculate \( 8^2 \).
Write the final answer as a simplified fraction.

Try solving on your own before revealing the answer!

Final Answer:

\( 8^{-2} = \frac{1}{64} \)

Because \( 8^2 = 64 \), so the reciprocal is \( \frac{1}{64} \).

Q9. Multiply the polynomials using the FOIL method: \( (4x - 6)(x + 12) \). Express the answer as a single polynomial in standard form.

Background

Topic: Multiplying Binomials (FOIL Method)

This question tests your ability to multiply two binomials using the FOIL method and combine like terms.

Key Terms and Formulas:

FOIL Method: First, Outer, Inner, Last terms multiplication.
Standard Form: Write the polynomial in descending order of exponents.

Step-by-Step Guidance

Multiply the First terms: \( 4x \times x \).
Multiply the Outer terms: \( 4x \times 12 \).
Multiply the Inner terms: \( -6 \times x \).
Multiply the Last terms: \( -6 \times 12 \).
Add all the terms together and combine like terms.

Try solving on your own before revealing the answer!

Final Answer:

\( (4x - 6)(x + 12) = 4x^2 + 42x - 72 \)

We used the FOIL method and combined like terms to write the answer in standard form.

Q10. Add the polynomials and express as a single polynomial in standard form: \( (x^2 - 2x + 8) + (3x^2 + x - 6) \).

Background

Topic: Adding Polynomials

This question tests your ability to add like terms and write the result in standard form.

Key Terms and Formulas:

Like Terms: Terms with the same variable and exponent.
Standard Form: Write the polynomial in descending order of exponents.

Step-by-Step Guidance

Write both polynomials in standard form.
Add the coefficients of like terms: \( x^2 \) terms, \( x \) terms, and constants.
Combine the results to write the final polynomial in standard form.

Try solving on your own before revealing the answer!

Final Answer:

\( (x^2 - 2x + 8) + (3x^2 + x - 6) = 4x^2 - x + 2 \)

We added like terms for each degree and wrote the result in standard form.

Q11. Use synthetic division to find the quotient and remainder when \( x^3 + 3x^2 - 3x + 7 \) is divided by \( x - 3 \).

Background

Topic: Synthetic Division

This question tests your ability to use synthetic division to divide a cubic polynomial by a linear factor and find both the quotient and remainder.

Key Terms and Formulas:

Synthetic Division: A shortcut method for dividing a polynomial by a binomial of the form \( x - c \).
Quotient and Remainder: The result of the division is \( Q(x) + \frac{R}{x-c} \).

Step-by-Step Guidance

Set up synthetic division using the zero of the divisor (for \( x - 3 \), use 3).
List the coefficients of the dividend: 1, 3, -3, 7.
Carry down the first coefficient, multiply by the zero, and add to the next coefficient. Repeat for all coefficients.
The last number is the remainder; the other numbers are the coefficients of the quotient.

Try solving on your own before revealing the answer!

Final Answer:

Quotient: \( x^2 + 6x + 15 \)

Remainder: 52

We used synthetic division with 3 and the coefficients 1, 3, -3, 7.

Q12. Factor the difference of two squares: \( 25x^2 - 9 \).

Background

Topic: Factoring the Difference of Squares

This question tests your ability to recognize and factor expressions of the form \( a^2 - b^2 \).

Key Terms and Formulas:

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

Step-by-Step Guidance

Recognize that \( 25x^2 \) is \( (5x)^2 \) and 9 is \( 3^2 \).
Apply the difference of squares formula: \( (a - b)(a + b) \).
Write the factors using the identified values for \( a \) and \( b \).

Try solving on your own before revealing the answer!

Final Answer:

\( 25x^2 - 9 = (5x - 3)(5x + 3) \)

We used the difference of squares formula with \( a = 5x \) and \( b = 3 \).