lesson
Updated 6 days ago · 2 views
If you have 3 dollar bills and 2 quarters, you have 5 items in your pocket — but you definitely don't have 5 dollar bills. In algebra, variables with different exponents are just as different as dollars and quarters.
Before we combine anything, let's look at the parts of an algebraic term. A coefficient is the numerical multiplier in front of a variable, while the exponent tells you how many times the variable multiplies by itself.
Be mindful of implicit values: a variable term without an explicit coefficient has a coefficient of 1 (e.g., x2=1x2, and −y3=−1y3). A variable with no written exponent has an implicit power of 1 (e.g., x=x1).
Be mindful of implicit values: a variable term without a written coefficient has an understood coefficient of 1 (such as x2=1x2 and −y3=−1y3). Likewise, a variable with no written exponent has an implicit power of 1 (x=x1).
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Like terms are terms that share the exact same variable letters raised to the exact same exponents. If the powers don't match identically, the terms are completely different types of quantities.
Why are we allowed to just add the numbers in front of like terms? Let's look at what is happening under the hood.
Why Combining Works
Collecting like terms is the direct structural inverse of the distributive law: a(xn)+b(xn)=(a+b)xn. For example, in 3x2+5x2, factoring out the common factor of x2 yields (3+5)x2=8x2.
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Because you are only counting how many x2 units you have, the exponent never changes when adding or subtracting.
What happens when an expression contains a long jumble of positive signs, negative signs, and different powers?