Polynomial Operations and Factoring

Example: Binomial Multiplication

$$\begin{aligned} (2x + 3)(x - 5) &= 2x(x) + 2x(-5) + 3(x) + 3(-5) \\ &= 2x^2 - 10x + 3x - 15 \\ &= 2x^2 - 7x - 15 \end{aligned}$$

Division

Polynomial division can be performed using long division or, in specific cases where the divisor is a linear expression (like $x - c$), through synthetic division. Long division is a more universal method that mimics the long division used in basic arithmetic.

Exam Tip: When performing polynomial long division, always ensure your dividend is written in descending order of exponents. If a term is "missing" (for example, if you have $x^3 + 1$ but no $x^2$ or $x$ term), use a placeholder like $0x^2$ and $0x$ to keep your columns aligned.

Factoring Polynomials

Factoring is the reverse of multiplication. It is the process of breaking down a polynomial into a product of simpler polynomials. This is a critical skill for finding the roots (zeros) of an equation.

Greatest Common Factor (GCF)

The first step in any factoring problem should always be to look for the Greatest Common Factor. This involves finding the largest number and the highest power of each variable that divides evenly into every term.

Example: Factoring by GCF

$$\begin{aligned} 6x^3 + 12x^2 &= 6x^2(x + 2) \end{aligned}$$

Factoring Trinomials

Trinomials are often in the form $ax^2 + bx + c$.

1. When $a = 1$: Look for two numbers that multiply to $c$ and add to $b$.
2. When $a \neq 1$: Use the "ac method." Multiply $a$ and $c$, then find two numbers that multiply to $ac$ and add to $b$. Split the middle term $bx$ into these two numbers and factor by grouping.

Example: Factoring $x^2 - 5x + 6$ We need two numbers that multiply to $6$ and add to $-5$. Those numbers are $-2$ and $-3$.

$$\begin{aligned} x^2 - 5x + 6 &= (x - 2)(x - 3) \end{aligned}$$

Special Products

Recognizing specific patterns can save significant time during exams:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$

Pro-Tip: Always check your factors by multiplying them back together. If you don't get the original polynomial, you've made a mistake in your signs or your arithmetic.

Summary Table of Operations

By mastering these fundamental operations and factoring techniques, you build the necessary foundation for calculus, physics, and advanced algebraic analysis.