3.6 Zeros of polynomial functions  (Page 7/14)

Key concepts

• To find $f(k),$ determine the remainder of the polynomial $f(x)$ when it is divided by $x-k.$ See [link] .
• $k$ is a zero of $f(x)$ if and only if $(x-k)$ is a factor of $f(x).$ See [link] .
• Each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See [link] and [link] .
• When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
• Synthetic division can be used to find the zeros of a polynomial function. See [link] .
• According to the Fundamental Theorem, every polynomial function has at least one complex zero. See [link] .
• Every polynomial function with degree greater than 0 has at least one complex zero.
• Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form $(x-c),$ where $c$ is a complex number. See [link] .
• The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
• The number of negative real zeros of a polynomial function is either the number of sign changes of $f(-x)$ or less than the number of sign changes by an even integer. See [link] .
• Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See [link] .

Section exercises

Verbal

Algebraic

For the following exercises, use the Remainder Theorem to find the remainder.

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.