College Algebra: Factoring and Simplifying Expressions Study Guide

Q11. Factor completely: $x^5 - 49x$

Background

Topic: Factoring Polynomials

This question tests your ability to factor out the greatest common factor (GCF) and then factor the difference of squares.

Key Terms and Formulas

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

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

Step-by-Step Guidance

1. Identify and factor out the GCF from $x^5 - 49x$.

2. After factoring out the GCF, observe if the remaining expression is a difference of squares.

3. Apply the difference of squares formula to factor further, if possible.

4. Write the fully factored form, but stop before multiplying out or simplifying the final factors.

Try solving on your own before revealing the answer!

Final Answer:

$x(x^4 - 49) = x(x^2 - 7)(x^2 + 7)$

We factored out $x$ and then used the difference of squares on $x^4 - 49$.

Q12. Factor completely: $3x^2 + 6x - 45$

Background

Topic: Factoring Quadratic Expressions

This question tests your ability to factor a quadratic trinomial by first factoring out the GCF and then factoring the quadratic.

Key Terms and Formulas

• Quadratic Trinomial: An expression of the form $ax^2 + bx + c$.

• Factoring by Grouping: Splitting the middle term to factor by grouping, if needed.

Step-by-Step Guidance

1. Identify and factor out the GCF from $3x^2 + 6x - 45$.

2. Write the remaining quadratic expression inside the parentheses.

3. Factor the quadratic expression, looking for two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

4. Set up the factors, but do not multiply out or simplify the final answer yet.

Try solving on your own before revealing the answer!

Final Answer:

$3(x^2 + 2x - 15) = 3(x - 3)(x + 5)$

We factored out $3$ and then factored the quadratic trinomial.

Q13. Factor $4x^3 - 27$

Background

Topic: Factoring Sums and Differences 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. Rewrite $4x^3 - 27$ as $(2x)^3 - 3^3$ to identify $a$ and $b$.

2. Apply the difference of cubes formula using $a = 2x$ and $b = 3$.

3. Write out the binomial and trinomial factors according to the formula.

4. Stop before expanding or simplifying the trinomial factor.

Try solving on your own before revealing the answer!

Final Answer:

$(2x - 3)(4x^2 + 6x + 9)$

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

Q14. Simplify: $(12x^3)(2x^2)$

Background

Topic: Multiplying Monomials

This question tests your ability to multiply coefficients and add exponents when multiplying like bases.

Key Terms and Formulas

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

Step-by-Step Guidance

1. Multiply the coefficients: $12 \times 2$.

2. Add the exponents of $x$: $x^3 \times x^2 = x^{3+2}$.

3. Combine the results to write the simplified expression, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Final Answer:

$24x^5$

We multiplied the coefficients and added the exponents for $x$.

Q15. Factor $10(2a-3)^2 - 19(2a-3) - 15$

Background

Topic: Factoring Quadratic Expressions in Disguise

This question tests your ability to use substitution to factor a quadratic expression where the variable is a binomial.

Key Terms and Formulas

• Substitution: Let $u = 2a - 3$ to rewrite the expression as a quadratic in $u$.

• Factoring Quadratics: $au^2 + bu + c$ can be factored into $(pu + q)(ru + s)$ if possible.

Step-by-Step Guidance

1. Let $u = 2a - 3$ and rewrite the expression in terms of $u$.

2. Factor the quadratic $10u^2 - 19u - 15$ by finding two numbers that multiply to $10 \times (-15)$ and add to $-19$.

3. Write the factored form in terms of $u$.

4. Substitute back $u = 2a - 3$ into your factors, but do not expand or simplify further.

Try solving on your own before revealing the answer!

Final Answer:

$[2(4a-11)(5a+3)]$

We used substitution, factored the quadratic, and then substituted back for $a$.