College Algebra Review: Simplifying, Combining, and Factoring Expressions

Q1. What is the simplest form of the expression $(2x^8)(3x^{-1})^2$? Write your answer in the form $ax^b$.

Background

Topic: Exponent Rules and Simplifying Expressions

This question tests your ability to apply exponent rules (such as the power of a product and power of a power) to simplify algebraic expressions.

Key Terms and Formulas

• Product of powers: $a^m \cdot a^n = a^{m+n}$

• Power of a power: $(a^m)^n = a^{mn}$

• Power of a product: $(ab)^n = a^n b^n$

Step-by-Step Guidance

1. Start by expanding $(3x^{-1})^2$ using the power of a product rule: $(3x^{-1})^2 = 3^2 \cdot (x^{-1})^2$.

2. Simplify $3^2$ and $(x^{-1})^2$ separately.

3. Multiply the result by $2x^8$ using the product of powers rule for $x$ terms.

4. Combine the coefficients and exponents to write the expression in the form $ax^b$.

Try solving on your own before revealing the answer!

Final Answer: $18x^6$

We expanded $(3x^{-1})^2$ to $9x^{-2}$, then multiplied by $2x^8$ to get $18x^6$.

Q2. Write an expression to represent the combined inventory of the two stores.

Background

Topic: Adding Polynomials

This question tests your ability to add polynomials by combining like terms, including fractional coefficients.

Key Terms and Formulas

• Polynomial: An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.

• Like terms: Terms that have the same variable raised to the same power.

Step-by-Step Guidance

1. Write out the polynomials for Store A and Store B: $\frac{1}{2}q^2 + \frac{7}{2}$ and $3q^2 - \frac{4}{5}q + \frac{1}{4}$.

2. Identify like terms in both polynomials (e.g., $q^2$ terms, $q$ terms, and constants).

3. Add the coefficients of like terms together, being careful with fractions.

4. Write the combined expression, grouping like terms.

Try solving on your own before revealing the answer!

Final Answer: $\frac{7}{2} + \frac{1}{4} + 3q^2 + \frac{1}{2}q^2 - \frac{4}{5}q$

Combine like terms to get the final polynomial representing the total inventory.

Q3. Which expression is equivalent to $210d^2 - 63d$?

Background

Topic: Factoring Polynomials

This question tests your ability to factor out the greatest common factor (GCF) from a polynomial expression.

Key Terms and Formulas

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

• Factoring: Rewriting an expression as a product of its factors.

Step-by-Step Guidance

1. Identify the GCF of the coefficients (210 and 63) and the variable part ($d^2$ and $d$).

2. Factor the GCF out of each term: $210d^2$ and $-63d$.

3. Write the expression as the product of the GCF and the remaining terms inside parentheses.

4. Compare your factored expression to the answer choices to find the equivalent form.

Try solving on your own before revealing the answer!

Final Answer: F. $21d(10d - 3)$

The GCF is $21d$, and factoring gives $21d(10d - 3)$.