Adding and Subtracting Polynomials

Operations with polynomials and combining like terms

What is a Polynomial?

A polynomial is a sum of terms with variables and exponents (whole numbers only).

Examples:

Like terms have the same variables with the same exponents.

Like terms: $3x^2$ and $-5x^2$ NOT like terms: $3x^2$ and $3x$ (different exponents)

Combine like terms: $(3x^2 + 2x - 1) + (x^2 - 5x + 4)$ $= 3x^2 + x^2 + 2x - 5x - 1 + 4$ $= 4x^2 - 3x + 3$

Distribute the negative sign, then combine like terms: $(5x^2 + 3x) - (2x^2 + x - 4)$ $= 5x^2 + 3x - 2x^2 - x + 4$ $= 3x^2 + 2x + 4$

Important: Change the sign of every term in the second polynomial!

Add: $(4x + 3) + (2x - 5)$

💡 Show Solution

Combine like terms:

$4x + 3 + 2x - 5$ $= (4x + 2x) + (3 - 5)$ $= 6x - 2$

Answer: $6x - 2$

Subtract: $(3x^2 + 5x - 2) - (x^2 - 3x + 4)$

💡 Show Solution

Step 1: Distribute the negative sign $3x^2 + 5x - 2 - x^2 + 3x - 4$

Step 2: Combine like terms $= (3x^2 - x^2) + (5x + 3x) + (-2 - 4)$ $= 2x^2 + 8x - 6$

Answer: $2x^2 + 8x - 6$

Simplify: $(2x^3 - x^2 + 4) + (x^3 + 3x^2 - 5) - (x^3 - 2x^2 + 1)$

💡 Show Solution

Step 1: Remove parentheses (distribute negative for subtraction) $2x^3 - x^2 + 4 + x^3 + 3x^2 - 5 - x^3 + 2x^2 - 1$

Step 2: Group like terms $= (2x^3 + x^3 - x^3) + (-x^2 + 3x^2 + 2x^2) + (4 - 5 - 1)$

Step 3: Combine $= 2x^3 + 4x^2 - 2$

Answer: $2x^3 + 4x^2 - 2$

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