Card 1 of 20
EasyStudy card 1
Prompt
What is the remainder when \(f(x)\) is divided by \(x-c\)?
Answer
\(f(c)\).
MathChapter 13: Polynomial and Radical Functions
Flashcards
Remainder Theorem and Factor Theorem Flashcards
Twenty concise cards on divisor signs, remainders, factors, zeros, intercepts, and parameter reasoning.
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Session progress0 of 20 studied
Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
For divisor \(x+3\), which input is used?
Answer
\(c=-3\), because \(x+3=x-(-3)\).
Card 3 of 20
EasyStudy card 3
Prompt
When is \(x-c\) a factor of \(f(x)\)?
Answer
Exactly when \(f(c)=0\).
Card 4 of 20
EasyStudy card 4
Prompt
If \(f(5)=7\), what is the remainder on division by \(x-5\)?
Answer
\(7\).
Card 5 of 20
EasyStudy card 5
Prompt
If \(f(-2)=0\), which binomial is a factor?
Answer
\(x+2\).
Card 6 of 20
EasyStudy card 6
Prompt
What is the key difference between a remainder and a factor test?
Answer
A factor requires the remainder to equal zero.
Card 7 of 20
EasyStudy card 7
Prompt
If \(f(c)=0\), what root is known?
Answer
\(x=c\).
Card 8 of 20
MediumStudy card 8
Prompt
If \(f(c)=0\), what intercept is known?
Answer
\((c,0)\).
Card 9 of 20
MediumStudy card 9
Prompt
What four ideas are equivalent when \(f(c)=0\)?
Answer
Zero/root \(c\), solution of \(f(x)=0\), intercept \((c,0)\), and factor \(x-c\).
Card 10 of 20
MediumStudy card 10
Prompt
Must you perform polynomial long division to find a linear remainder?
Answer
No. Substitute the matching value into the polynomial.
Card 11 of 20
MediumStudy card 11
Prompt
What should you do before applying either theorem?
Answer
Rewrite the divisor in the form \(x-c\).
Card 12 of 20
MediumStudy card 12
Prompt
If \(f(4)=-6\), is \(x-4\) a factor?
Answer
No; its remainder is \(-6\), not zero.
Card 13 of 20
MediumStudy card 13
Prompt
If a table contains \((3,0)\), what factor follows?
Answer
\(x-3\).
Card 14 of 20
MediumStudy card 14
Prompt
If a graph crosses at \((-1,0)\), what factor follows?
Answer
\(x+1\).
Card 15 of 20
MediumStudy card 15
Prompt
How can a parameter be found so \(x-c\) is a factor?
Answer
Set \(f(c)=0\) and solve for the parameter.
Card 16 of 20
MediumStudy card 16
Prompt
Does a nonzero remainder identify an \(x\)-intercept?
Answer
No. An intercept requires a function value of zero.
Card 17 of 20
HardStudy card 17
Prompt
If \(f(c)=2\) and \(g(c)=-1\), what is the remainder of \(3f+4g\) at \(x-c\)?
Answer
\(3(2)+4(-1)=2\).
Card 18 of 20
HardStudy card 18
Prompt
Why is the sign in \(x-c\) a common trap?
Answer
The substitution value is the opposite of the displayed constant when the divisor is written \(x+k\).
Card 19 of 20
HardStudy card 19
Prompt
How can divisibility of \(af(x)+bg(x)\) by \(x-c\) be tested?
Answer
Evaluate \(af(c)+bg(c)\); divisibility holds if the result is zero.
Card 20 of 20
HardStudy card 20
Prompt
If \(f(c)=0\), does that determine the degree of \(f\)?
Answer
No. It identifies a root and factor but does not determine degree.
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