Factoring Examples Calculator

Common Factoring Methods:

\[ \begin{align*} &\text{1. GCF: } ax + ay = a(x + y) \\ &\text{2. Difference of Squares: } a^2 - b^2 = (a+b)(a-b) \\ &\text{3. Perfect Square Trinomial: } a^2 \pm 2ab + b^2 = (a \pm b)^2 \\ &\text{4. Trinomial: } x^2 + bx + c = (x + m)(x + n) \\ &\text{5. Sum/Difference of Cubes: } a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) \end{align*} \]

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1. What is Factoring?

Factoring is the process of breaking down an expression into simpler parts (factors) that when multiplied together give the original expression. It's a fundamental skill in algebra used to simplify expressions and solve equations.

2. Common Factoring Methods

The main factoring techniques include:

\[ \begin{align*} &\text{• Greatest Common Factor (GCF)} \\ &\text{• Difference of Squares } a^2 - b^2 = (a+b)(a-b) \\ &\text{• Perfect Square Trinomials} \\ &\text{• General Trinomials} \\ &\text{• Sum/Difference of Cubes} \end{align*} \]

Examples:

\( x^2 - 9 \) factors to \( (x+3)(x-3) \)
\( x^2 + 6x + 9 \) factors to \( (x+3)^2 \)
\( 2x^2 + 7x + 3 \) factors to \( (2x+1)(x+3) \)

3. Importance of Factoring

Details: Factoring is essential for solving quadratic equations, simplifying rational expressions, finding roots of polynomials, and in calculus for limit problems.

4. Using the Calculator

Tips: Enter polynomial expressions using ^ for exponents (e.g., x^2 for x squared). The calculator demonstrates common factoring patterns.

5. Frequently Asked Questions (FAQ)

Q1: What's the first step in factoring?
A: Always look for a Greatest Common Factor (GCF) first before trying other methods.

Q2: How do you factor trinomials?
A: For \( x^2 + bx + c \), find two numbers that multiply to c and add to b.

Q3: What's the difference of squares formula?
A: \( a^2 - b^2 = (a+b)(a-b) \) - used when you have two perfect squares separated by subtraction.

Q4: Can all polynomials be factored?
A: All polynomials can be factored, but some require complex numbers. Some trinomials are "prime" over integers.

Q5: Why is factoring important in real life?
A: Factoring is used in engineering, physics, computer science (cryptography), economics, and many other fields.