Zeros of Polynomials

Front

Define the zeros of a polynomial.

Back

The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. They are also known as the roots of the polynomial.

Front

What is the relationship between the degree of a polynomial and the number of zeros it can have?

Back

A polynomial of degree n can have at most n zeros, counting multiplicities.

Front

How do you find the zeros of a polynomial given one zero?

Back

If one zero is known, you can use polynomial division to divide the polynomial by the corresponding linear factor, then find the zeros of the resulting polynomial.

Front

What is the Factor Theorem?

Back

The Factor Theorem states that if a polynomial f(x) has a zero at x = a, then (x - a) is a factor of f(x).

Front

What is the Remainder Theorem?

Back

The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a).

Front

Find the zeros of the polynomial: f(x) = (x - 2)(x + 3).

Back

The zeros are x = 2 and x = -3.

Front

If a polynomial has a zero at x = 1, what can you say about the polynomial's value at x = 1?

Back

The value of the polynomial at x = 1 is zero, i.e., f(1) = 0.

Front

What is the significance of complex zeros in polynomials?

Back

Complex zeros occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points.

Front

How do you determine the multiplicity of a zero?

Back

The multiplicity of a zero is determined by the number of times the corresponding factor appears in the polynomial.

Front

Find the zeros of the polynomial: f(x) = x^2 - 5x + 6.

Back

The zeros are x = 2 and x = 3.