MAT 171 Unit 4 Review 3

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, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Front

What does the Complex Conjugate Root Theorem state?

Back

The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number a + bi is a root, then its complex conjugate a - bi is also a root.

Front

If a + bi is a root of a polynomial, what is the other root?

Back

The other root is a - bi.

Front

How do you find a polynomial function with given zeros?

Back

To find a polynomial function with given zeros, use the fact that if r is a root, then (x - r) is a factor. Multiply the factors corresponding to all the roots to get the polynomial.

Front

What is the least degree of a polynomial?

Back

The least degree of a polynomial is the smallest exponent of the variable in the polynomial when it is expressed in standard form.

Front

What are integral coefficients in a polynomial?

Back

Integral coefficients are the numerical factors in a polynomial that are whole numbers (positive, negative, or zero).

Front

Write a polynomial function of least degree with integral coefficients that has the zeros -1, 5, and 2i.

Back

The polynomial is (x + 1)(x - 5)(x - 2i)(x + 2i) = (x + 1)(x - 5)(x^2 + 4) = x^4 - 4x^3 - 19x^2 + 20x + 20.

Front

What is the significance of complex roots in polynomials?

Back

Complex roots indicate that the polynomial does not intersect the x-axis at those points, and they occur in conjugate pairs when the polynomial has real coefficients.

Front

How do you determine the degree of a polynomial?

Back

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

Front

What is a zero of a polynomial?

Back

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