Flashcards: Polynomial Functions and Theorems — 22 cards

What is the general form of a polynomial expression in x of degree n?

It is $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ with $a_n \neq 0$.

What must the degree n be in a polynomial expression?

A nonnegative integer.

What condition must the leading coefficient $a_n$ satisfy in a polynomial?

It must not be zero.

What exponents or variable placements are not allowed in polynomial functions?

Polynomial functions cannot have negative or fractional exponents, variables in denominators, inside radicals, or decimal exponents.

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

Because it contains a negative exponent.

Why is $2x^{1/2}+3x+5$ not a polynomial?

Because it contains a fractional exponent.

What makes an expression with a variable in the denominator or inside a radical not a polynomial?

Variables in denominators or inside radicals disqualify it as a polynomial function.

What does the Rational Root Theorem state about rational roots of integer-coefficient polynomials?

Any rational root in lowest terms has numerator dividing the constant term and denominator dividing the leading coefficient.

What form do possible rational zeros take for $f(x)=a_nx^n+\cdots+a_1x+a_0$?

They have the form $\frac{p}{q}$ where p divides the constant term and q divides the leading coefficient.

What is the first step to apply the Rational Root Theorem?

Identify the constant term of the polynomial.

After identifying coefficients, what is the next step in applying the Rational Root Theorem?

List the factors of the constant term and the leading coefficient.

How do you form possible rational roots using the Rational Root Theorem?

Form all possible values of $\frac{p}{q}$ from factors of constant term and leading coefficient.

What does the Polynomial Remainder Theorem state about dividing by x−c?

The remainder equals f(c).

How do you apply the Remainder Theorem to a polynomial?

Substitute the candidate value c into the polynomial and evaluate f(c).

What is the value of f(1) for $f(x)=x^3-6x^2+11x-6$?

f(1) equals 0.

What does the Factor Theorem state about x−c and f(x)?

x−c is a factor of f(x) if and only if f(c)=0.

What factor corresponds to a positive zero 1?

The factor x−1 corresponds to the zero 1.

What factor corresponds to a negative zero −1?

The factor x+1 corresponds to the zero −1.

Which values give zero remainder for $f(x)=x^3-6x^2+11x-6$?

The values 1, 2, and 3 give zero remainder.

What are the zeros of $f(x)=x^3-6x^2+11x-6$?

The zeros are x=1, x=2, and x=3.

Are x+1, x+2, x+3 factors of $f(x)=x^3-6x^2+11x-6$?

No, because f(−1), f(−2), and f(−3) are not zero.

Are x−6 and x+6 factors of $f(x)=x^3-6x^2+11x-6$?

No, because f(6) and f(−6) are not zero.