2.1.3 · D5Algebra — Introduction & Intermediate
Question bank — Addition and subtraction of algebraic expressions
This is a thinking bank for the parent topic. No heavy arithmetic here — every item targets a misconception or a boundary case. Cover the reveal, commit to an answer with a reason, then check.
First — what does a "term" even look like?
Before the traps, ground the picture. A term is a number (the coefficient) glued onto a variable part. Think of the variable as a type of box, and the coefficient as how many boxes you have.
Look at the figure: three green boxes labelled make ; five pink boxes labelled make . You can count more green boxes, but you can never turn a green box into a pink one — that is the whole reason and can't merge into one term.
Now watch what "combining like terms" looks like when you slide matching boxes together, and what the Distributive Property "pull-out" does:
The distributive step is literally gathering all the -boxes into one pile and counting them — the box type is factored out front, and only the counts and add.
True or false — justify
can be simplified to .
False — and are different box types (apples and oranges), so there is no common factor to pull out; is already fully simplified.
and are like terms because they share the same variable .
False — like terms need the same variable and the same power; , so they stay separate.
and are like terms.
True — multiplication is commutative, so and are the identical variable part; they combine to .
For any expression, .
True — subtracting is the same as adding the opposite, which is exactly why the minus in front of a bracket flips every inside sign (multiply by ).
and are like terms because they use the same letters.
False — the powers differ ( vs ), so the variable parts are different objects; they do not combine.
The constant term and the constant term are like terms.
True — both have no variable (a variable part of ), so they combine to .
Adding gives .
False — addition only combines coefficients; exponents stay put, giving . Adding exponents is a multiplication rule.
Removing parentheses preceded by a sign never changes any inside sign.
True — a leading is multiply-by-, so every term keeps its sign.
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True — zero of anything is nothing, so both terms vanish and the whole expression collapses to the constant .
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False — counts two of the same box, giving ; means , which is multiplication, not addition.
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True — coefficients can be decimals or fractions; only the counts add () while the box type stays the same.
Spot the error
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The minus was only applied to , not to the ; correct is .
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A number and a variable term are unlike ( has no ), so they can't merge; the answer stays .
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Only the coefficients change; the variable part is untouched, giving , not stacked-up exponents.
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Only the first term was flipped; multiplying the whole bracket by gives — the middle and last signs also flip.
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Fractional coefficients still just add: , so the answer is — you never add the denominators.
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Identical terms with opposite signs cancel: , not .
then "stuck".
The like terms weren't grouped: and , giving simply .
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The inner bracket was cleared but the outer minus wasn't fully distributed; correctly .
Nested parentheses — full distributive walk-through
Why questions
Why can we combine but not ?
Both and share the box type , so the Distributive Property lets us pull it out as ; and share no common factor, so nothing can be pulled out.
Why does the coefficient change during addition while the variable part stays the same?
Adding like terms is counting how many boxes you have; the box (the variable part) is unchanged — you only update the count in front.
Why must the minus sign hit every term inside a bracket?
A leading minus means "multiply the whole bracket by ", and by the distributive property multiplies each term individually, flipping every sign.
Why do exponents stay fixed when adding but grow when multiplying?
Adding counts boxes of the same power → still power ; multiplying stacks the multiplications ( used times) → power .
Why is "adding the opposite" a safer way to do subtraction of expressions?
Rewriting as forces you to distribute the negative once, up front, so you never forget an inside sign later.
Why can a coefficient be a fraction or decimal but the counting picture still works?
You can have "two-and-a-half boxes" — a partial count is still a count, so merges its counts to exactly like whole numbers do.
Edge cases
Is a "like-terms" problem, and what is the result?
Yes — same box type; combining gives , a degenerate case where the term disappears entirely.
What is ?
The term contributes nothing, so the expression is just the constant ; a zero coefficient erases its term.
Are and different terms?
No — a variable with no visible coefficient has an understood coefficient of , so and are literally the same term.
Can an expression like be simplified further?
No — all three terms have different variable parts (, , ), so there are no like terms to combine; it is already simplest. See Polynomials.
If you subtract an expression from itself, e.g. , what happens?
Every term cancels its twin: and , leaving — the additive-identity edge case.
Is a variable term or a constant?
A constant — for any nonzero , so a term like is really just the number and combines with other constants. (Order matters here; see Order of Operations.)
Does the order you write like terms in the final answer matter, e.g. vs ?
No — addition is commutative so both are equal; convention just usually writes higher-degree or alphabetically-first terms first for tidiness.
Can a coefficient like ever become a whole-number term?
Yes — fractional counts can sum to a whole: , giving exactly .
Recall One-line survival summary
Only combine terms with the same variable and the same power; when you do, only the coefficients change (whole, fraction, or decimal); a leading minus flips every sign inside its bracket, worked deepest-bracket first.