Quiz: Polynomial Functions and Theorems — 11 questions

Question 1

1. Which condition is required for an expression to have the standard polynomial form of degree $n$ ?

Its variable may appear in a denominator if the coefficients are real

Its exponents are nonnegative integers and its coefficients are real

Its exponents may include fractional values and its coefficients are integers

Its exponents may include negative integers and its coefficients are real

Explanation

A polynomial uses nonnegative integer exponents, real coefficients, and a nonzero leading coefficient. Allowing negative or fractional exponents would produce a non-polynomial expression.

Answer

Its exponents are nonnegative integers and its coefficients are real

Question 2

2. A polynomial has highest nonzero term $-4x^7$ . What is its degree and leading coefficient?

Degree

-4

and leading coefficient

7

Degree

7

and leading coefficient

4

Degree

7

and leading coefficient

-4

Degree

-4

and leading coefficient

-7

Explanation

The degree is the exponent of the highest-degree nonzero term, while the leading coefficient is its numerical coefficient. Thus, $-4x^7$ gives degree $7$ and leading coefficient $-4$ .

Answer

Degree $$7$$ and leading coefficient $$-4$$

Question 3

3. Which expression is a polynomial function?

\frac{3}{x^4}-2x+7

3x^4-2x+7

3x^{1/2}-2x+7

3x^{-4}-2x+7

Explanation

The first expression has only nonnegative integer exponents and no variable in a denominator or radical. The other expressions contain a negative exponent, a fractional exponent, or a variable in the denominator.

Answer

$$3x^4-2x+7$$

Question 4

4. Why is $2x^{-2}+3x+5$ not a polynomial?

It contains a negative exponent

It contains a linear term

It contains real coefficients

It contains a nonzero constant term

Explanation

The term $2x^{-2}$ has a negative exponent, which is not permitted in a polynomial. A nonzero constant term, a linear term, and real coefficients are all compatible with polynomial expressions.

Answer

It contains a negative exponent

Question 5

5. For a polynomial with integer coefficients, what does the Rational Root Theorem require of a rational root written as $\frac{p}{q}$ in lowest terms?

Both

p

and

q

divide the constant term

Both

p

and

q

divide the leading coefficient

p

divides the constant term and

q

divides the leading coefficient

p

divides the leading coefficient and

q

divides the constant term

Explanation

The theorem identifies possible rational roots by taking the numerator from factors of the constant term and the denominator from factors of the leading coefficient. It produces candidates rather than proving that every candidate is an actual root.

Answer

$$p$$ divides the constant term and $$q$$ divides the leading coefficient

Question 6

6. For $f(x)=6x^3-5x^2+4x-12$ , which set contains all possible rational zeros generated by the Rational Root Theorem?

\pm1,\pm2,\pm3,\pm6,\pm12,\pm\frac14,\pm\frac15,\pm\frac16

\pm1,\pm2,\pm4,\pm6,\pm12,\pm\frac15,\pm\frac25,\pm\frac35

\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac16,\pm\frac13,\pm\frac23

\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac12,\pm\frac13,\pm\frac23

Explanation

The constant term is $-12$ and the leading coefficient is $6$ , so candidates have numerators from factors of $12$ and denominators from factors of $6$ . The listed set includes the resulting reduced fractions and integer candidates; the other sets omit valid combinations or use factors from the wrong coefficient.

Answer

$$\pm1,\pm2,\pm3,\pm4,\pm6,\pm12,\pm\frac16,\pm\frac13,\pm\frac23$$

Question 7

7. When a polynomial $f(x)$ is divided by $x-c$ , what does the Polynomial Remainder Theorem identify as the remainder?

The value of the constant term

The quotient evaluated at

c

The coefficient of the highest-degree term

The value

f(c)

Explanation

The theorem states that evaluating the polynomial at the divisor’s root, $c$ , gives the remainder. The quotient is a separate polynomial and is not generally equal to $f(c)$ .

Answer

The value $$f(c)$$

Question 8

8. What procedure finds the remainder when a polynomial is divided by $x-c$ using the Remainder Theorem?

Replace each occurrence of

x

with

x-c

and simplify

Substitute

c

into the polynomial and evaluate the result

Factor the polynomial before identifying its constant term

Divide every coefficient by

c

and add the quotients

Explanation

The required process is direct substitution of $c$ for $x$ , followed by evaluation of $f(c)$ . Polynomial division is not needed to obtain the remainder through this theorem.

Answer

Substitute $$c$$ into the polynomial and evaluate the result

Question 9

9. According to the Factor Theorem, when is $x-c$ a factor of $f(x)$ ?

When

f(c)=0

When

f(c)

equals the quotient

When

f(c)=1

When

f(c)

has the highest degree

Explanation

The Factor Theorem says that $x-c$ is a factor precisely when evaluating the polynomial at $c$ gives zero. A nonzero value of $f(c)$ indicates that the proposed linear expression is not a factor.

Answer

When $$f(c)=0$$

Question 10

10. Which factor corresponds to the zero $-1$ of a polynomial?

x-1

x^2+1

x+1

-x+1

Explanation

A zero $c$ corresponds to the factor $x-c$ , so substituting $c=-1$ produces $x-(-1)=x+1$ . The factor $x-1$ instead corresponds to the positive zero $1$ .

Answer

$$1, 2, 3$$

Question 11

11. For $f(x)=x^3-6x^2+11x-6$ , which set contains all three tested values that produce a remainder of zero?

-3, 3, 6

1, 2, 6

1, 2, 3

-1, -2, -3

Explanation

Evaluating the polynomial at $1$ , $2$ , and $3$ gives zero, so these values are zeros and their corresponding linear factors divide the polynomial. The other sets include values whose evaluations are nonzero.

Quiz: Polynomial Functions and Theorems — 11 questions

Detailed questions and answers

1. Which condition is required for an expression to have the standard polynomial form of degree $n$ ?

2. A polynomial has highest nonzero term $-4x^7$ . What is its degree and leading coefficient?

3. Which expression is a polynomial function?

4. Why is $2x^{-2}+3x+5$ not a polynomial?

5. For a polynomial with integer coefficients, what does the Rational Root Theorem require of a rational root written as $\frac{p}{q}$ in lowest terms?

6. For $f(x)=6x^3-5x^2+4x-12$ , which set contains all possible rational zeros generated by the Rational Root Theorem?

7. When a polynomial $f(x)$ is divided by $x-c$ , what does the Polynomial Remainder Theorem identify as the remainder?

8. What procedure finds the remainder when a polynomial is divided by $x-c$ using the Remainder Theorem?

9. According to the Factor Theorem, when is $x-c$ a factor of $f(x)$ ?

10. Which factor corresponds to the zero $-1$ of a polynomial?

11. For $f(x)=x^3-6x^2+11x-6$ , which set contains all three tested values that produce a remainder of zero?

Review with flashcards

Read the study sheet

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Quiz: Polynomial Functions and Theorems — 11 questions

Detailed questions and answers

1. Which condition is required for an expression to have the standard polynomial form of degree nnn?

2. A polynomial has highest nonzero term −4x7-4x^7−4x7. What is its degree and leading coefficient?

3. Which expression is a polynomial function?

4. Why is 2x−2+3x+52x^{-2}+3x+52x−2+3x+5 not a polynomial?

5. For a polynomial with integer coefficients, what does the Rational Root Theorem require of a rational root written as pq\frac{p}{q}qp​ in lowest terms?

6. For f(x)=6x3−5x2+4x−12f(x)=6x^3-5x^2+4x-12f(x)=6x3−5x2+4x−12, which set contains all possible rational zeros generated by the Rational Root Theorem?

7. When a polynomial f(x)f(x)f(x) is divided by x−cx-cx−c, what does the Polynomial Remainder Theorem identify as the remainder?

8. What procedure finds the remainder when a polynomial is divided by x−cx-cx−c using the Remainder Theorem?

9. According to the Factor Theorem, when is x−cx-cx−c a factor of f(x)f(x)f(x)?

10. Which factor corresponds to the zero −1-1−1 of a polynomial?

11. For f(x)=x3−6x2+11x−6f(x)=x^3-6x^2+11x-6f(x)=x3−6x2+11x−6, which set contains all three tested values that produce a remainder of zero?

Review with flashcards

Read the study sheet

Similar courses

Group and Vector Space Algebra

Photosynthesis and Respiration

Types of Chemical Reactions

Ray Optics and Optical Instruments

Sustainability Transition Foundations

Statistics for Management

Create your own quizzes

1. Which condition is required for an expression to have the standard polynomial form of degree $n$ ?

2. A polynomial has highest nonzero term $-4x^7$ . What is its degree and leading coefficient?

4. Why is $2x^{-2}+3x+5$ not a polynomial?

5. For a polynomial with integer coefficients, what does the Rational Root Theorem require of a rational root written as $\frac{p}{q}$ in lowest terms?

6. For $f(x)=6x^3-5x^2+4x-12$ , which set contains all possible rational zeros generated by the Rational Root Theorem?

7. When a polynomial $f(x)$ is divided by $x-c$ , what does the Polynomial Remainder Theorem identify as the remainder?

8. What procedure finds the remainder when a polynomial is divided by $x-c$ using the Remainder Theorem?

9. According to the Factor Theorem, when is $x-c$ a factor of $f(x)$ ?

10. Which factor corresponds to the zero $-1$ of a polynomial?

11. For $f(x)=x^3-6x^2+11x-6$ , which set contains all three tested values that produce a remainder of zero?