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Factoring

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Factoring

Study Guide

Key Definition

Factoring is the process of breaking down an expression into a product of simpler expressions or numbers, such as $(x + 3)(x + 5)$.

Important Notes

To factor $x^2 + bx + c$, find two numbers that multiply to $c$ and add to $b$.
If no such numbers exist, the polynomial is irreducible.
Use the distributive property to simplify factored expressions.
Irreducible polynomials cannot be factored into simpler polynomials.
Practice helps in recognizing patterns in factoring.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$x^2$ represents $x$ squared

Remember to use proper notation when solving problems

Why It Works

Factoring works by reversing the expansion of polynomials. For example, $(n+3)(n+5)$ expands to $n^2 + 8n + 15$, and factoring reverses this process.

Remember

To factor $x^2 + bx + c$, look for pairs of factors of $c$ that sum to $b$.

Quick Reference

Distributive Property:$a(b + c) = ab + ac$

Factoring Formula:$x^2 + bx + c = (x + m)(x + n)$ where $m + n = b$ and $mn = c$

Understanding Factoring

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Step 1: List all factor pairs of the constant term (15) for $x^2 + 8x + 15$. Step 2: Identify the pair that sums to 8 (3 and 5). Step 3: Write the factored form as $(x + 3)(x + 5)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the factored form of $n^2 + 8n + 15$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are organizing a school event and need to set up rectangular tables. The total area can be represented by $x^2 + 8x + 15$. How would you factor this to determine the dimensions?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Is $x^2 + 37x + 100$ factorable? If not, explain why.

Challenge Quiz

Single Choice Quiz

Advanced

What is the factored form of $n^2 + 29n + 100$?

Recap

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Review key concepts and takeaways