Unit 10 · Lesson 10.3

Polynomials — terms, degree, standard form; add & subtract

A polynomial is just a sum of terms like 3x² + 2x − 1: number-times-a-power pieces, added up. It's the noun the next three units are about, so this lesson gives the parts their names and then does the two easiest things you can do with them: add and subtract.

The adding is nothing new; it's Unit 2's "combine like terms" with fresh vocabulary. The subtracting hides one genuine trap, and most of the lesson's care goes there.

First, the vocabulary, all on one example. Take 5x² − 3x + 2. It has three pieces, so it's a trinomial. Its highest power is 2, so its degree is 2. The number riding the highest-power term is its leading coefficient, here 5. And it's already in standard form, meaning its terms run from highest power down to lowest. A useful way to see why the plain number sits last: think of 2 as 2x⁰, since x⁰ = 1. That makes it the lowest-degree term, so it naturally ends the line.

Now adding. Like terms have to match the variable and its power: 3x² and x² are both "x²-pieces," so they combine, while 3x² and 3x are different kinds and don't. Picture the x²-pieces and the x-pieces as two different kinds of box: you can only stack a box with its own kind. Add the coefficients within each kind and keep them sorted by degree: $$(3x^2+2x-1)+(x^2-5x+4) = (3+1)x^2+(2-5)x+(-1+4) = 4x^2-3x+3$$ Nothing fancy: x²-pieces with x²-pieces, x-pieces with x-pieces, plain numbers with plain numbers.

Subtraction is the same idea with one step that's easy to fumble, so go slow here. Subtracting a whole polynomial means a −1 shakes hands with every term inside the second parentheses: every sign flips, not just the first. This is the exact "hand the minus to everyone" move from Unit 2, wearing new clothes: $$(2x^2+3x)-(x^2-x+2) = 2x^2+3x \;\underbrace{-x^2+x-2}_{\text{every sign flipped}} = x^2+4x-2$$ Watch the two that bite: the −x inside became +x, and the +2 became −2. The first term flipping is obvious; it's these later ones that get left alone by accident.

Here's the move that catches it for you: put a simple number in for x and check both sides. At x = 1, the original is (2 + 3) − (1 − 1 + 2) = 5 − 2 = 3, and the result is 1 + 4 − 2 = 3. They match, so the signs are right.

If they hadn't matched, that wouldn't be a failure. Your check would have just caught a slipped sign before it cost you anything. The next move would be to go back and re-run the flips one term at a time, because a mismatch here is almost always one sign left unflipped, not a broken method.