Polynomial Operations

Polynomial operations involve adding, subtracting, and multiplying algebraic expressions that contain multiple terms. Mastering these operations is a fundamental skill in algebra.

Adding and Subtracting Polynomials

To add or subtract polynomials, you simply combine like terms. Like terms are terms that have the exact same variables raised to the exact same powers.

Addition Example: $(2x^2 + 3x - 5) + (x^2 - 2x + 4)$ Group the like terms together: $= (2x^2 + x^2) + (3x - 2x) + (-5 + 4)$ $= 3x^2 + x - 1$

Subtraction Example: When subtracting, remember to distribute the negative sign to every term in the second polynomial. $(4x^2 - 2x) - (x^2 + 5x)$ $= 4x^2 - 2x - x^2 - 5x$ Combine like terms: $= 3x^2 - 7x$

Multiplying Polynomials

Multiplying polynomials requires using the distributive property. Every term in the first polynomial must be multiplied by every term in the second polynomial.

Multiplying Binomials (The FOIL Method)

When multiplying two binomials, use the FOIL method to keep track of your multiplication:

• First terms
• Outer terms
• Inner terms
• Last terms

Example: Expand $(3x + 2)(x - 5)$

• First: $3x \cdot x = 3x^2$
• Outer: $3x \cdot (-5) = -15x$
• Inner: $2 \cdot x = 2x$
• Last: $2 \cdot (-5) = -10$

Combine them and simplify the like terms: $3x^2 - 15x + 2x - 10$ $= 3x^2 - 13x - 10$

Special Product Patterns

Sometimes you will encounter special patterns, such as squaring a binomial. The pattern for a perfect square binomial $(a - b)^2$ is $a^2 - 2ab + b^2$.

Example: Expand $(2x - 3)^2$ Using the pattern where $a = 2x$ and $b = 3$:

• $a^2 = (2x)^2 = 4x^2$
• $-2ab = -2(2x)(3) = -12x$
• $b^2 = (-3)^2 = 9$

Putting it all together: $4x^2 - 12x + 9$

(Tip: If you forget the pattern, you can always write it out as $(2x - 3)(2x - 3)$ and use the FOIL method to get the exact same result!)