5.5 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 $x-k.$ This is known as the Remainder Theorem. See [link] .
• According to the Factor Theorem, $k$ is a zero of $f(x)$ if and only if $(x-k)$ is a factor of $f(x).$ See [link] .
• According to the Rational Zero Theorem, 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.