Begin here
A polynomial is a math expression made of pieces added together. The pieces are called terms. This room is about reading one: seeing what terms it is made of, and putting the same kinds of term together. That is most of what the GED test asks you to do with a polynomial.
It begins with a short story about the man who wrote the first algebra book. Then it goes one small step at a time. At the end, a practice page lets you test yourself.
Tap Read at the top of any page to hear it read aloud. The voice comes from your own phone or computer; you can choose it under Contents.
Squares, things, and coins
In Baghdad, around the year 820, the scholar al-Khwarizmi wrote the book that gave algebra its name. He wrote every problem in words. For the number he was looking for, he wrote the thing. For that number multiplied by itself, the square, he wrote wealth, the word for a person’s property. And plain numbers he counted in dirhams, the silver coin of Baghdad.
شيءshay, the thing مالmal, wealth درهمdirham, a coin
So one of his problems reads: a wealth and ten things equal thirty-nine dirhams. Today we write it x2 + 10x = 39. Three kinds of piece: a square, some things, and some coins.
He never added a thing to a coin. Ten things and five coins stayed ten things and five coins. That one rule, keep each kind separate and count each kind on its own, is what this room teaches. When you have it, you can read any polynomial.
What a polynomial is
An expression is a piece of math with no equals sign in it. It does not say that two things are equal; it just describes an amount. 3x + 7 is an expression. 3x + 7 = 19 is not; it is an equation, and an equation can be solved.
A polynomial is an expression made of pieces added together. The pieces are called terms. A term can be a plain number, like 7. It can be a letter, like x. It can be a number times a letter, like 3x, or a number times a letter with an exponent, like 5x2. The exponent is the small raised number. x2 means x times x, and x3 means x times x times x.
Poly means many, so a polynomial is an expression with many terms. But the name is loose. One term counts, and so do two. Tap each of these to see its terms.
A polynomial with one term is also called a monomial, with two terms a binomial, and with three terms a trinomial. The GED test uses these names now and then. They only count the terms.
What is not a polynomial
Three things keep an expression from being a polynomial. Each one is a letter in a place a polynomial does not allow.
- A letter under a division line. 5x means 5 divided by x. Dividing by a letter is not allowed. Dividing by a number is fine: x5 is a polynomial, because it is x divided by 5, which is one fifth of x.
- A letter under a root sign. √x, the square root of x, is not allowed.
- An exponent that is not a whole number. The exponents in a polynomial are 0, 1, 2, 3, and so on. A negative exponent like x−2 is not allowed, and neither is a fraction. The letter cannot be the exponent either: 2x is not a polynomial.
Sort these. For each one, say whether it is a polynomial.
Cut it into terms
To read a polynomial, first cut it into its terms. The cuts go in front of the plus and minus signs. Each sign stays with the term to its right; the sign is part of that term.
Take 3x2 − 5x + 7. Cut in front of the minus and in front of the plus, and you have three terms: 3x2, −5x, and +7. The middle term is negative five x. The minus belongs to it.
The first term has no sign in front. A term with no sign is positive. Its plus sign is there; it is just not written.
The parts of a term
Look inside one term: −5x2. The number in front is the coefficient. It counts how many: −5x2 is negative five of x2. The letter is the variable, the thing whose value we do not know. The small raised number is the exponent. It tells how many times the variable is multiplied by itself.
Two of these hide. When a term has no number in front, the coefficient is 1: x means 1x. When a variable has no exponent showing, the exponent is 1: x means x1.
A term with no variable at all, like +7, is called a constant. It stays the same whatever x is.
The shape of a term
Here is a way to picture the three kinds of term you will meet most. A plain number is counted in small squares: 7 is seven small squares. Think of al-Khwarizmi’s coins. The variable x is a strip. Its length is x, the number we do not know, so the strip is longer than a small square, but nobody can say by how much. And x2 is x times x: a big square, x long and x wide.
A term is a pile of one shape. 3x2 is three big squares. 5x is five strips. 7 is seven small squares. The coefficient says how many; the variable and the exponent say which shape. Tap a term to see its pile.
x3 would be a cube, x long, x wide, and x tall. After that we run out of shapes to draw, but the rule holds: the exponent tells the kind.
Degree, and the usual order
The degree of a term is its exponent. 5x2 has degree 2. 5x has degree 1, because x is x1. A plain number has no variable, so its degree is 0.
The degree of a whole polynomial is the biggest degree in it. In 3x + 2x2 − 7, the biggest exponent is 2, so the polynomial has degree 2. The 3 and the 7 do not come into it. The degree comes from the exponents, never from the coefficients.
A polynomial is usually written with the biggest exponent first, then the next biggest, down to the plain number: 2x2 + 3x − 7. This order is called standard form. It puts the degree in front where you can see it, and each sign travels with its term when the terms change places.
Like terms
Two terms are like terms when they have the same variable with the same exponent. In tiles, they are the same shape. 3x and 5x are like terms: both strips. 3x and 3x2 are not: a strip and a big square. 5 and 5x are not: a small square and a strip.
The coefficient does not matter for this. 100x and −2x are like terms. What matters is the shape: the variable and its exponent.
Like terms can be counted together into one term, because they are the same kind of thing. Three strips and five strips are eight strips: 3x + 5x = 8x. Unlike terms cannot. 3x + 5 stays 3x + 5, three strips and five small squares, and there is nothing more to do with it.
Put like with like
Now the whole move. Take 2x2 + 4x − 1 + 5x + x2. It looks like a lot. Read it in four steps, and it becomes 3x2 + 9x − 1.
Cut it into terms, each sign with its term. Color the kinds: big squares one color, strips another, small squares a third. Then put like with like: gather each kind together, in the usual order, the sign moving with its term. Finally, count each pile. That is all there is to it: sorting, then counting.
On the GED test this is called combining like terms, or simplifying the expression. Simplifying means writing it with as few terms as possible.
Counting with holes
What about minus? A negative term is drawn as a hole: an empty place the shape of the tile. −3x is three strip-shaped holes.
A tile and a hole of the same shape cancel each other. The tile fills the hole, and nothing is left. So 3x − 3x is 0: three strips fill three holes.
5x − 3x is five strips and three holes. Three strips fill the three holes, and two strips remain: 5x − 3x = 2x. Now turn it around. 3x − 5x is three strips and five holes. Three holes get filled, and two holes remain: 3x − 5x = −2x. The answer is negative because holes were left over.
In symbols, you count the coefficients with their signs: 3 − 5 = −2, so the answer is −2x. Two holes and three more holes are five holes: −2x − 3x = −5x. Tap an expression to watch the tiles.
The sign travels with its term
When you gather like terms, each sign moves with its term. In 3x − 5 + 2x, the 5 is negative. Put like with like and it is 3x + 2x − 5. The minus came along, and the answer is 5x − 5.
The common mistake is to leave the sign behind and write 3x + 2x + 5. Then the answer comes out 5x + 5 instead of 5x − 5, and it is wrong by ten.
Underlining each term with its sign, the way this room does, is a habit that stops the mistake. On paper, circle each term with its sign before you move anything. Then whatever moves, moves whole.
You decide
Your turn. Each problem has two or three answers. The wrong ones are traps, and the traps have names. Pick one, and the page tells you why it is right, or what the trap was. Wrong picks cost nothing: try again. On the sorting problems, you can watch the sort after you answer.
Where the road leads
You can now read a polynomial: cut it into terms, name the parts of a term, find the degree, and put like with like. Squares with squares, things with things, coins with coins, as al-Khwarizmi did.
The next room is about parentheses. It asks what 2(x + 3) means, and what happens to the signs when you subtract a whole expression in parentheses, as in (4x + 1) − (x − 5). The same tiles and the same holes will do the work.
Where this history comes from
- Al-Khwarizmi’s book is al-Kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala, “The Short Book on Calculation by Restoring and Balancing,” written in Baghdad around the year 820. The English translation we lean on is Frederic Rosen’s, published in London in 1831.
- In the book, the unknown number is called shay (thing) and also jidhr (root). Its square is called mal (wealth, or property). Plain numbers are counted in dirhams, the silver coin of the time. The problem quoted here reads, in Rosen’s translation, “a square and ten roots are equal to thirty-nine dirhems.”
- The word polynomial is built from Greek poly, many, and the ending of binomial, which comes from a Latin word for a name. A binomial is an expression with two named parts.
- Writing an exponent as a small raised number is a habit that spread from René Descartes’s La Géométrie (1637), the same book that made x the usual name for the unknown.