College Algebra Study Guide: Factoring and Simplifying Expressions

Q11. Factor completely: $x^3 - 49x$

Background

Topic: Factoring Polynomials

This question tests your ability to factor a cubic polynomial by recognizing common factors and using the difference of squares.

Key Terms and Formulas:

• Common factor: A term that can be factored out from all terms in the expression.

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

Step-by-Step Guidance

1. Identify the common factor in both terms. Here, $x$ is a common factor.

2. Factor $x$ out: $x^3 - 49x = x(x^2 - 49)$

3. Notice that $x^2 - 49$ is a difference of squares.

4. Apply the difference of squares formula to $x^2 - 49$.

Try solving on your own before revealing the answer!

Final Answer: $x(x-7)(x+7)$

Factoring $x^2 - 49$ as $(x-7)(x+7)$ gives the complete factorization.

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

Background

Topic: Factoring Quadratic Expressions

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

Key Terms and Formulas:

• Greatest common factor (GCF): The largest factor shared by all terms.

• Factoring trinomials: $ax^2 + bx + c$ can often be factored as $(mx + n)(px + q)$.

Step-by-Step Guidance

1. Identify the GCF in $3x^2 + 6x - 45$. The GCF is $3$.

2. Factor out $3$: $3(x^2 + 2x - 15)$

3. Now, factor the quadratic $x^2 + 2x - 15$.

4. Look for two numbers that multiply to $-15$ and add to $2$.

Try solving on your own before revealing the answer!

Final Answer: $3(x+5)(x-3)$

The numbers $5$ and $-3$ satisfy the conditions, so the trinomial factors as $(x+5)(x-3)$.

Q13. Factor $4x^3 - 27$

Background

Topic: Factoring Cubic Expressions

This question tests your ability to factor a 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. Recognize that $4x^3$ and $27$ are both perfect cubes: $4x^3 = (2x)^3$, $27 = 3^3$.

2. Rewrite the expression as $(2x)^3 - 3^3$.

3. Apply the difference of cubes formula.

4. Set up the factors: $(2x - 3)$ and $(2x)^2 + (2x)(3) + 3^2$.

Try solving on your own before revealing the answer!

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

Using the formula, the cubic factors as shown.

Q14. Simplify: $(12x^3)(2x^2)(4x^4)$

Background

Topic: Multiplying Monomials

This question tests your ability to multiply coefficients and add exponents when multiplying powers of the same base.

Key Terms and Formulas:

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

• Multiply coefficients: $12 \cdot 2 \cdot 4$

Step-by-Step Guidance

1. Multiply the coefficients: $12 \cdot 2 \cdot 4$

2. Add the exponents for $x$: $x^3 \cdot x^2 \cdot x^4 = x^{3+2+4}$

3. Combine the results to write the simplified expression.

Try solving on your own before revealing the answer!

Final Answer: $24x^9$

Multiplying the coefficients gives $24$, and adding the exponents gives $x^9$.

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

Background

Topic: Factoring Quadratic Expressions in Disguise

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

Key Terms and Formulas:

• Let $u = 2a - 3$ to rewrite the expression as $10u^2 - 19u - 15$.

• Factoring trinomials: $au^2 + bu + c$ can often be factored as $(mu + n)(pu + q)$.

Step-by-Step Guidance

1. Let $u = 2a - 3$ and rewrite the expression as $10u^2 - 19u - 15$.

2. Factor the quadratic $10u^2 - 19u - 15$.

3. Look for two numbers that multiply to $10 \cdot (-15) = -150$ and add to $-19$.

4. Split the middle term and factor by grouping.

Try solving on your own before revealing the answer!

Final Answer: $(2a-5)(5(2a-3)+3)$

Factoring gives two binomial factors in terms of $a$.