Factoring Trinomials in College Algebra

Q1. Factor the trinomial: $x^2 + 14x + 45$

Background

Topic: Factoring Trinomials

This question tests your ability to factor quadratic expressions of the form $ax^2 + bx + c$ where $a = 1$. Factoring is a key skill in college algebra, especially for solving quadratic equations and simplifying expressions.

Key Terms and Formulas

• Trinomial: A polynomial with three terms.

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

• Standard Form: $ax^2 + bx + c$

Step-by-Step Guidance

1. Check for a Greatest Common Factor (GCF) among all terms. If there is one, factor it out first.

2. Identify $a$, $b$, and $c$ in the trinomial. Here, $a = 1$, $b = 14$, $c = 45$.

3. Look for two numbers that multiply to $c$ (45) and add to $b$ (14).

4. Write the trinomial as a product of two binomials: $(x + m)(x + n)$, where $m$ and $n$ are the numbers found in the previous step.

Try solving on your own before revealing the answer!

Q2. Factor the trinomial: $n^2 - 3n - 28$

Background

Topic: Factoring Trinomials

This question asks you to factor a quadratic expression where $a = 1$ and $c$ is negative. This often means one factor will be positive and one negative.

Key Terms and Formulas

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

• Factoring: Expressing as a product of binomials.

Step-by-Step Guidance

1. Check for a GCF among all terms. If there is one, factor it out first.

2. Identify $a = 1$, $b = -3$, $c = -28$.

3. Find two numbers that multiply to $-28$ and add to $-3$.

4. Write the trinomial as $(n + m)(n + n)$, using the numbers found.

Try solving on your own before revealing the answer!

Q3. Factor the trinomial: $2a^2 + 2a - 24$

Background

Topic: Factoring Trinomials with a GCF

This question involves factoring out a GCF before factoring the remaining trinomial.

Key Terms and Formulas

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

• Factoring: Expressing as a product of binomials.

Step-by-Step Guidance

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

2. Write the remaining trinomial after factoring out the GCF.

3. Factor the resulting trinomial using the methods above.

Try solving on your own before revealing the answer!

Q4. Factor the trinomial: $4p^2 - 24p + 20$

Background

Topic: Factoring Trinomials with a GCF

This question also involves factoring out a GCF before factoring the remaining trinomial.

Key Terms and Formulas

• GCF: The largest factor shared by all terms.

• Factoring: Expressing as a product of binomials.

Step-by-Step Guidance

1. Find and factor out the GCF from all terms.

2. Write the remaining trinomial after factoring out the GCF.

3. Factor the resulting trinomial using the standard method.

Try solving on your own before revealing the answer!