Adding, Subtracting, and Multiplying Polynomials

Think of a polynomial as a stack of coins sorted by denomination: all the x² terms go in one column, all the x terms in another, the plain numbers in a third. Adding or subtracting two polynomials just means combining matching columns — x² with x², x with x, constant with constant. You never mix an x² coin with an x coin, because they measure different things. Subtraction has one extra step: subtracting a polynomial means flipping the sign of every term inside it, not just the first one, because the minus sign is really multiplying the whole group by -1.

Multiplication works differently: every term in the first polynomial has to shake hands with every term in the second. With (2x-3) and (x²+4x-5), the 2x multiplies all three terms of the second polynomial, then the -3 does the same, giving six small products. Only after that full distribution do you sort the results into columns by power of x and add the coefficients in each column. Skipping terms during distribution is the most common way to lose a piece of the answer.

This column-and-distribute idea is the same engine that runs factoring, just in reverse, and it is what makes later work with quadratics and higher-degree polynomials manageable. Once a polynomial is in standard form, with powers listed from highest to lowest, it is ready for finding roots, graphing, or matching against another expression — all tasks that assume the terms have already been correctly combined.