112 Part 2: Into the Unknown
WORLDLY WISDOM
What is the pattern to x + x^2?
If x = 1, x + x^2 = 2 or 2x.
If x = 2, x + x^2 = 6 or 3x.
If x = 3, x + x^2 = 12 or 4x.
Can you see it? The answer is always a multiple of x, and the multiplier is 1 more than
the value of x. You can write it like this: x + x^2 = (x + 1)x. This is called the factored
form of x + x^2. If you use the distributive property to multiply x(x + 1), you’ll see that it
equals x + x^2.Think of it this way: x stands for a number, which we could imagine as the length of a line seg-
ment. If x is the length of a line segment, then x^2 would be the area of a square whose sides are x
units long. Visually, a line segment and a square are very different things. Even though both x and
x^2 have an x in their names, they’re very different things, different numbers. They’re unlike terms.Even though they both contain the same variable, x and x^2 represent different ideas. There’s
something going on in x^2 that’s not happening in the plain x. Visually, you can think of the x^2
expanding into a square, while the x is still a line segment. Numerically, the x^2 has multiplication
going on that’s not happening in the simple x. The x and the x^2 are clearly related, but they’re not
the same. They’re not like terms.
Different variables are clearly not alike, but even terms built from the same variable may be dif-
ferent from one another. To be like terms, terms must have the exact same variable and the exact
same exponent. Their variable parts are exact matches. Only their coefficients are different. That
means 4y^3 and -7y^3 are like, because they have the same variable and the same exponent. The fact
that the 4 and the -7 are different coefficients is okay. The coefficients are just telling you how
many you have.DEFINITION
Like terms are terms that have the same variable, raised to the same power. For
example, the terms 4x and 7x are like terms.x x^2xx