Combining Terms and Polynomial Operations

Objective: Combine coefficients of like powers.

Prerequisite skills: signed arithmetic and distributive property. A polynomial is a sum of terms with nonnegative integer powers of its variables. Like terms have identical variable parts, so only their coefficients are added. Constants form their own group. Common error: adding exponents when adding terms. Restrictions and interpretation: Combine terms only when their variable factors and exponents match.

Objective: Use distribution to find a quadratic coefficient.

Multiply each term of one polynomial by every term of the other, then combine like powers. The middle coefficient of (x + a)(x + b) is a + b and the constant is ab. Common error: using ab as the coefficient of x instead of the constant. Restrictions and interpretation: Combine terms only when their variable factors and exponents match.

Objective: Preserve signs when subtracting one whole expression from another.

Subtraction of a polynomial distributes −1 to every term of the subtrahend. After changing those signs, combine like terms. Evaluating at a test input can check the resulting expression. Common error: changing only the first sign inside a subtracted group. Restrictions and interpretation: Combine terms only when their variable factors and exponents match. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.