Roots of Polynomials

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

Front

What are 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

How do you find the zeros of the polynomial f(x) = (x + 2)(x - 3)?

Back

Set f(x) = 0: (x + 2)(x - 3) = 0. The zeros are x = -2 and x = 3.

Front

What does the multiplicity of a root mean?

Back

The multiplicity of a root refers to the number of times a particular root appears in the factorization of a polynomial. A root with an even multiplicity will touch the x-axis, while a root with an odd multiplicity will cross the x-axis.

Front

Write a polynomial function with zeros at 1, -2, and 3 (all multiplicity of 1).

Back

f(x) = (x - 1)(x + 2)(x - 3).

Front

How many roots does the polynomial y = x^3 - 6x^2 + 11x - 6 have?

Back

It has 3 roots, as the degree of the polynomial is 3.

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. For example, in f(x) = 2x^4 + 3x^2 + 1, the degree is 4.

Front

Given the polynomial y = x^2 - 4, how many roots does it have?

Back

It has 2 roots, which can be found by factoring: y = (x - 2)(x + 2).

Front

What is the factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its factors. For example, f(x) = x^2 - 5x + 6 can be factored as f(x) = (x - 2)(x - 3).

Front

How do you determine the number of real roots of a polynomial?

Back

You can determine the number of real roots by using the discriminant (b^2 - 4ac for quadratic polynomials) or by analyzing the polynomial's degree and its behavior.