5.5-5.8 Zeros of Polynomial Functions

Front

What does the Rational Root Theorem state?

Back

The Rational Root Theorem states that any rational solution (or root) of a polynomial equation, in the form of p/q, is such that p is a factor of the constant term and q is a factor of the leading coefficient.

Front

List the possible rational roots for the polynomial 2x^3 - 11x^2 + 12x + 9 = 0.

Back

±1, ±3, ±9, ±1/2, ±3/2, ±9/2.

Front

How do you find all possible rational zeros of a polynomial?

Back

To find all possible rational zeros, use the Rational Root Theorem to list factors of the constant term and leading coefficient, then form all possible p/q combinations.

Front

What are the possible rational zeros for f(x) = 3x^5 - 5x^2 + x + 6?

Back

1/3, 2/3, 1, 2, 3, 6, -1/3, -2/3, -1, -2, -3, -6.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression.

Front

True or False: The degree of the polynomial determines the number of roots.

Back

True.

Front

How do you factor the polynomial P(x) = x^3 - 3x^2 - 5x + 15?

Back

The factors are (x - 3)(x - √5)(x + √5).

Front

What is a zero of a polynomial function?

Back

A zero of a polynomial function is a value of x that makes the function equal to zero.

Front

What is synthetic division and when is it used?

Back

Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor.

Front

How can you verify if a number is a zero of a polynomial?

Back

Substitute the number into the polynomial. If the result is zero, then it is a zero of the polynomial.