02/10 Polynomial Functions Free Online FlashCards

What is a polynomial function?

A polynomial function has the form f(x)=aₙxⁿ+…+a₁x+a₀, where the exponents are nonnegative integers and the coefficients are constants.

How are degree and leading coefficient identified?

The degree is the greatest exponent with a nonzero coefficient. The leading coefficient is the coefficient of that highest-degree term.

How are polynomials added or subtracted?

Combine terms only when they have the same variable and exponent. For subtraction, distribute the negative sign before combining.

Evaluate f(2) for f(x)=2x³−x²+4x−1.

Substitute 2 for x: f(2)=2(2)³−(2)²+4(2)−1=27.

What does the Factor Theorem state?

f(c)=0 exactly when x−c is a factor of f(x). Thus, every zero c produces a corresponding linear factor x−c.

How does multiplicity affect an x-intercept?

An odd-multiplicity zero makes the graph cross the x-axis; an even-multiplicity zero makes it touch the axis and turn around.

What is the polynomial Division Algorithm?

Polynomial division has the form f(x)=d(x)q(x)+r(x), where r=0 or the degree of r is less than the degree of d.

Which number is used for synthetic division by x+2?

Synthetic division uses the number c when dividing by x−c. For x+2=x−(−2), the synthetic-division number is −2.

What does the Remainder Theorem determine?

When f(x) is divided by x−c, the remainder is f(c). A zero remainder means x−c is a factor.

What does the Rational Zero Theorem provide?

If p/q is a rational zero in lowest terms, p must divide the constant term and q must divide the leading coefficient. The theorem lists possibilities, not guaranteed zeros.

What are the zeros of x³−6x²+11x−6?

x³−6x²+11x−6=(x−1)(x−2)(x−3), so the zeros are x=1, 2, and 3.

What determines a polynomial’s end behavior?

Even degree with positive leading coefficient: both ends rise. Even negative: both fall. Odd positive: left falls and right rises. Odd negative: left rises and right falls.