Card 1 of 20
EasyStudy card 1
Prompt
What separates terms in a polynomial?
Answer
Addition or subtraction signs at the top grouping level.
Example
\(4x^2-3x+7\) has three terms.
MathChapter 10: Law of Exponents and Polynomials
Flashcards
Adding, Subtracting, Multiplying, and Dividing Polynomials Flashcards
Recall polynomial vocabulary, degree, operations, and long-division checks.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What is a coefficient?
Answer
The numerical factor multiplying a variable part.
Example
In \(-8x^3\), the coefficient is \(-8\).
Card 3 of 20
EasyStudy card 3
Prompt
How do you find the degree of a monomial with several variables?
Answer
Add the exponents of all its variable factors.
Example
\(5x^2y^4\) has degree \(6\).
Card 4 of 20
EasyStudy card 4
Prompt
How do you find the degree of a simplified polynomial?
Answer
Take the greatest degree among its nonzero terms.
Card 5 of 20
EasyStudy card 5
Prompt
What makes two terms like terms?
Answer
They have exactly the same variables raised to exactly the same powers.
Example
\(3x^2y\) and \(-7x^2y\) are like terms.
Card 6 of 20
MediumStudy card 6
Prompt
Can terms with the same total degree always be combined?
Answer
No. Their complete variable parts must match.
Example
\(x^2y\) and \(xy^2\) both have degree \(3\) but are unlike.
Card 7 of 20
EasyStudy card 7
Prompt
What is the safest first step when subtracting a polynomial?
Answer
Distribute the negative sign to every term of the polynomial being subtracted.
Card 8 of 20
EasyStudy card 8
Prompt
How are polynomials added?
Answer
Align like powers and add the coefficients in each matching column.
Card 9 of 20
MediumStudy card 9
Prompt
How do you multiply a polynomial by a monomial?
Answer
Distribute the monomial to every term, multiplying coefficients and adding matching-base exponents.
Card 10 of 20
MediumStudy card 10
Prompt
How do you divide a polynomial by a monomial?
Answer
Divide every polynomial term by the monomial separately.
Card 11 of 20
MediumStudy card 11
Prompt
Why insert a zero coefficient for a missing degree before long division?
Answer
It preserves alignment of equal powers during multiplication and subtraction.
Example
Write \(x^3-5x+1\) as \(x^3+0x^2-5x+1\).
Card 12 of 20
MediumStudy card 12
Prompt
What repeated cycle drives polynomial long division?
Answer
Divide leading terms, multiply the divisor, subtract the full product, and bring down the next term.
Card 13 of 20
EasyStudy card 13
Prompt
What identity verifies polynomial division?
Answer
Dividend equals divisor times quotient plus remainder.
Formula
\[P(x)=D(x)Q(x)+R(x)\]
Card 14 of 20
MediumStudy card 14
Prompt
What degree condition must a nonzero polynomial remainder satisfy?
Answer
Its degree must be less than the divisor's degree.
Formula
\[\deg R<\deg D\]
Card 15 of 20
MediumStudy card 15
Prompt
In long division, what creates the next quotient term?
Answer
Divide the current remainder's leading term by the divisor's leading term.
Card 16 of 20
MediumStudy card 16
Prompt
What sign error commonly occurs in polynomial long division?
Answer
Subtracting only the first term instead of distributing subtraction across the entire product row.
Card 17 of 20
EasyStudy card 17
Prompt
What is the degree of a nonzero constant?
Answer
\(0\)
Card 18 of 20
EasyStudy card 18
Prompt
Simplify \(3x^2+5x^2\).
Answer
\(8x^2\)
Card 19 of 20
HardStudy card 19
Prompt
Why is evaluating at one value not a proof that two polynomials are equivalent?
Answer
Different polynomials can agree at a single input; symbolic coefficient comparison or expansion proves identity.
Card 20 of 20
HardStudy card 20
Prompt
What quick check catches many polynomial-operation errors?
Answer
Evaluate the original and result at several safe inputs, then use symbolic structure for the proof.
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