MathChapter 1: The Language and Tools of Algebra
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Prompt
What is a term?
💡Keep the sign with the term.
Answer
A part of an expression separated by addition or subtraction.
Example\(3x,-2y,7\) are three terms.
Prompt
What is a coefficient?
💡Include its sign.
Answer
The signed numerical factor multiplying a variable part.
ExampleIn \(-8x\), the coefficient is \(-8\).
Prompt
When are terms like terms?
💡Coefficients may differ.
Answer
When their variable parts, including exponents, are identical.
Example\(3x^2\) and \(-x^2\) are like.
Prompt
Can \(2x\) and \(3y\) combine?
💡Only identical variable parts combine.
Answer
No; their variable parts differ.
ExampleThe sum remains \(2x+3y\).
Prompt
Can \(x\) and \(x^2\) combine?
💡Exponent is part of the variable part.
Answer
No; their exponents differ.
ExampleThey remain \(x+x^2\).
Prompt
How do like terms combine?
💡Do not add exponents.
Answer
Add or subtract coefficients and keep the variable part unchanged.
Prompt
State the distributive property.
💡Every internal term receives the factor.
Answer
\(a(b+c)=ab+ac\).
Prompt
What is the negative-distribution rule?
💡Think \(-1\) times the grouping.
Answer
A negative factor changes the sign of every internal term.
Prompt
What does the commutative property change?
💡Move each term with its sign.
Answer
Order, not grouping or value.
Prompt
What does the associative property change?
💡Parentheses move.
Answer
Grouping, not order or value.
Example\((a+b)+c=a+(b+c)\).
Prompt
What is simplest form?
💡Scan for grouping and matching variable parts.
Answer
A form with all required distribution complete and no remaining like terms to combine.
Example\(5x-2\) is simpler than \(2x+3x-2\).
Prompt
Simplify \(4x-7x+2\).
💡Combine signed coefficients.
Answer
\(-3x+2\)
Prompt
Simplify \(-2(x-5)\).
💡Negative times negative is positive.
Answer
\(-2x+10\)
ExampleBoth internal terms receive \(-2\).
Prompt
How do fractional coefficients combine?
💡Treat coefficients as ordinary fractions.
Answer
Use a common denominator, combine them, and preserve the variable part.
Example\(x/2+3x/4=5x/4\).
Prompt
How should a subtracted expression be handled?
💡A minus affects the whole expression.
Answer
Keep it grouped, then distribute \(-1\) to every term.
Example\(5x-(2x+3)=3x-3\).
Prompt
Why is \((x+3)^2\ne x^2+9\)?
💡Distribution is multiplication over a sum.
Answer
An exponent does not distribute over addition; expansion also has the term \(6x\).
Example\((x+3)^2=x^2+6x+9\).
Prompt
How can a test value disprove equivalence?
💡Use \(0,1,-1\) first.
Answer
If the expressions differ for any one allowed value, they are not equivalent.
ExampleAt \(x=1\), \(2(x+3)=8\) but \(2x+3=5\).
Prompt
What does complete cancellation mean?
💡Do not force a variable into the answer.
Answer
Like terms sum to zero and disappear; a constant or zero may remain.
Example\(6x-2(3x-5)=10\).
Prompt
What is the reliable simplification order?
💡Do not combine through parentheses first.
Answer
Distribute, reorder signed terms, combine like terms, then audit.
ExampleFor \(3(x+2)+x\), distribute before combining.
Prompt
How do symbolic coefficients behave?
💡Letters can be coefficients too.
Answer
Like numerical coefficients: distribute and combine them algebraically.
Example\(a(x+2)+b(x-1)=(a+b)x+2a-b\).
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