Skip to main content
Contents Dark Mode Prev Up Next \(\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\renewcommand{\neg}{\sim}
\newcommand{\brac}[1]{\left( #1 \right)}
\newcommand{\rad}[1]{#1 \ \text{rad}}
\newcommand{\eval}[1]{\left. #1 \right|}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\ang}[1]{#1^\circ}
\newcommand{\crossmethod}[4]{
\begin{tikzpicture}[baseline=(M.base)]
\node (M) at (0,0) {$#1$};
\node (P) at (0,-1) {$#2$};
\node (N) at (1.5,0) {$#3$};
\node (Q) at (1.5,-1) {$#4$};
\draw (M) -- (Q);
\draw (P) -- (N);
\end{tikzpicture}
}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 9.3 Factoring and Roots of Polynomial Functions
Recall that our goal is to be able to factor polynomials, in a similar way that we can factor quadratic functions. The question is: how do we find factors?
Example 9.3.1 . Factoring a Quadratic.
Recall that quadratics can be written in factored form. For example,
\begin{align*}
f(x) \amp = x^2-x-6\\
f(x) \amp = (x+2)(x-3)
\end{align*}
The factors are
\(x+2\) and
\(x-3\text{.}\)
The zeros are
\(x = -2\) and
\(x = 3\text{.}\)
Notice that each zero corresponds to a factor. In other words,
\begin{gather*}
\text{Zero of the polynomial} \iff \text{Factor of the polynomial}
\end{gather*}
This relationship extends to any polynomial function, and is called the
factor theorem .
Subsection 9.3.1 Factor Theorem
Theorem 9.3.2 . Factor theorem.
Let \(f(x)\) be a polynomial. Then, \(x - a\) is a factor of the polynomial \(f(x)\) if and only if \(f(a) = 0\text{.}\) In other words,
If
\(f(a) = 0\text{,}\) then
\(x - a\) is a factor of
\(f\text{.}\)
If
\(x - a\) is a factor of
\(f\text{,}\) then
\(f(a) = 0\text{.}\)
This provides an equivalence between
\(x - a\) being a factor of
\(f(x)\text{,}\) and
\(a\) being a root of
\(f(x)\text{.}\) We also know that
\(x-a\) being a factor of
\(f(x)\) means that dividing
\(f(x)\) by
\(x-a\) gives a remainder of 0. Putting it together, these are all equivalent:
\begin{gather*}
\begin{pmatrix} f(x) \\ \text{has factor} \\ x - a \end{pmatrix} \iff \begin{pmatrix} f(x) \text{ has root $a$} \\ \text{or, } f(a) = 0 \end{pmatrix} \iff \begin{pmatrix} \text{$f(x)$ divided by $x - a$} \\ \text{gives remainder 0} \end{pmatrix}
\end{gather*}
Example 9.3.3 . Using the Factor Theorem.
For the polynomial function
\(f(x) = 2x^3 - 3x^2 - 10x + 3\text{,}\) to determine if
\(x-3\) is a factor, evaluate
\(f(3)\text{,}\)
\begin{align*}
f(3) \amp = 2(3)^3 - 3(3)^2 - 10(3) + 3\\
\amp = 0
\end{align*}
This means that
\(x=3\) is a root, so by the factor theorem,
\(x - 3\) is a factor of
\(f\text{.}\)
Exercise Group 9.3.1 . Zeros and Corresponding Factors.
For each zero of a polynomial function
\(f(x)\text{,}\) give the corresponding factor.
(a) \(f(1)=0\)
(b) \(f(-3)=0\)
(c) \(f(4)=0\)
(d) \(f(a)=0\)
Exercise Group 9.3.2 . Identifying Factors I.
For
\(f(x) = x^3 - x^2 - 5x + 2\text{,}\) determine if each binomial is a factor.
(a) \(x-1\)
(b) \(x+2\)
Exercise Group 9.3.3 . Identifying Factors II.
For
\(f(x) = x^4 + 6x^3 + 4x^2 + 7x + 60\text{,}\) determine if each binomial is a factor.
(a) \(x+5\)
(b) \(x-2\)
Exercise Group 9.3.4 . Finding Unknowns with the Factor Theorem.
Solve for the unknown variables using the factor theorem.
(a) \(k\) so that \(x+1\) is a factor of \(2x^4 + (k+1)x^2 - 6kx + 11\text{.}\)
(b) \(k\) for which \(\frac{1}{2}\) is a zero of \(P(x) = -4x^3 + 2x^2 - 2kx + k^3\text{?}\)
(c) \(x-a\) is a factor of \(2x^3 - a x^2 + (1-a^2)x + 5\text{,}\) what is \(a\text{?}\)
(d) \(a\) and \(b\) such that \(x-1\) is a factor of both \(x^3 + x^2 + ax + b\) and \(x^3 - x^2 - ax + b\text{.}\)
Subsection 9.3.2 The Remainder Theorem
The remainder theorem states that when a polynomial
\(f(x)\) is divided by a binomial
\(x - a\text{,}\) the remainder is
\(f(a)\text{.}\)
Theorem 9.3.4 . Remainder theorem.
If a polynomial
\(f(x)\) is divided by a binomial
\(x - a\text{,}\) the remainder is equal to the function evaluated at
\(a\text{,}\) i.e.
\(f(a)\text{.}\)
\begin{gather*}
\boxed{f(a) = \begin{matrix} \text{remainder of } p(x) \\ \text{divided by } x-a \end{matrix}}
\end{gather*}
In other words, to find the remainder after division by
\(x-a\text{,}\) you can just evaluate the polynomial at
\(x = a\text{,}\) instead of going through the division process.
Exercise Group 9.3.5 . Finding Remainders.
Find the remainder for each polynomial division, using the remainder theorem.
(a) \((x^2 + 5x + 4) \div (x-2)\)
(b) \((x^3 + 3x^2 - 5x + 2) \div (x+1)\)
(c) \((x^2 - 2x) \div (x-4)\)
(d) \((x^3 - 2x^2 - 3) \div (x+2)\)
(e) \((2x - 4x^3 - 3x^2) \div (x-2)\)
(f) \(\dfrac{x^3 + 2x^2 - 3x + 9}{x+3}\)
(g) \((x^3 + 2x^2 - 3x + 5) \div (x-3)\)
(h) \(\dfrac{x^4 - 3x^2 - 5x + 2}{x-2}\)
Exercise Group 9.3.6 . Solving for an Unknown Coefficient I.
Find each value of
\(m\) from the given division and remainder.
(a) \((mx^2 - 2x - 1) \div (x-1)\text{,}\) remainder \(-2\)
(b) \((x^3 - mx^2 + 17x + 6) \div (x-3)\text{,}\) remainder \(12\)
(c) \((x^4 - 2x^3 - x^2 + mx + 2) \div (2x-1)\text{,}\) remainder \(-1\)
(d) \((2x^4 - x^3 - 4x^2 + x - m) \div (x+3)\text{,}\) remainder \(3\)
Exercise Group 9.3.7 . Solving for an Unknown Coefficient II.
Find
\(k\) for each polynomial, using the given remainder condition.
(a) \(f(x) = 4x^3 + kx^2 - 7x + 9\) is divided by \(x+3\text{,}\) the remainder is \(-33\text{.}\)
(b) \((x^3 + 4x^2 - x + k) \div (x-1)\) has remainder \(3\text{.}\)
(c) \((x^3 + x^2 + kx - 15) \div (x-2)\) has remainder \(3\text{.}\)
(d) \((x^3 + kx^2 + x + 5) \div (x+2)\) has remainder \(3\text{.}\)
(e) \((kx^3 + 3x + 1) \div (x+2)\) has remainder \(3\text{.}\)
(f) \((3x^2 + 6x - 10) \div (x+k)\) has remainder \(14\text{.}\)
(g) \(x^3 + kx + 1\) is divided by \(x-2\text{,}\) the remainder is \(-3\text{.}\)
(h) \(x^3 - x^2 + kx - 8\) is divided by \(x-4\text{,}\) the remainder is \(0\text{.}\)
(i) \(2x^4 + kx^2 - 3x + 5\) is divided by \(x-2\text{,}\) the remainder is \(3\text{.}\)
(j) \(x^3 + kx + 6\) is divided by \(x+2\text{,}\) the remainder is \(4\text{.}\)
(k) \(p(x)=kx^{50} + 2x^{30} + 4x + 7\) is divided by \(x+1\text{,}\) the remainder is \(23\text{.}\) Find \(k\text{.}\)
Exercise Group 9.3.8 . Problems with Equal Remainders.
Solve the following problem.
(a) \(p(x)= -2x^3 + cx^2 - 5x + 2\) has the same remainder when divided by \(x-2\) and by \(x+1\text{.}\) Find \(c\text{.}\)
Exercise Group 9.3.9 . Solving for Multiple Unknown Coefficients.
Find the unknown coefficients for each polynomial.
(a) \(3x^3 - mx^2 + nx + 2\) has remainders \(-6\) upon division by \(x+1\) and \(18\) upon division by \(x-2\text{.}\) Find \(m\) and \(n\text{.}\)
(b) \(2x^3 + mx^2 + nx - 14\) has remainders \(22\) upon division by \(x-2\) and \(-53\) upon division by \(x+3\text{.}\) Find \(m\) and \(n\text{.}\)
(c) \(mx^3 - 3x^2 + nx + 2\) has remainders \(-1\) upon division by \(x+3\) and \(-4\) upon division by \(x-2\text{.}\) Find \(m\) and \(n\text{.}\) Answer .
\(m=-\frac{11}{5}, n=\frac{59}{5}\)
(d) \(3x^3 + ax^2 + bx - 9\) has remainders \(-5\) upon division by \(x-2\) and \(-16\) upon division by \(x+1\text{.}\) Find \(a,b\text{.}\) Answer .
\(a=-\frac{14}{3}, b=-\frac{2}{3}\)
(e) \(kx^3 + mx^2 + x - 2\) is divided by \(x-1\text{,}\) the remainder is \(6\text{.}\) When this polynomial is divided by \(x+2\text{,}\) the remainder is \(12\text{.}\) Find \(k\) and \(m\text{.}\)
(f) \(x^4 + kx^3 - mx + 15\) has no remainder when divided by \(x-1\) and \(x+3\text{.}\) Find \(k\) and \(m\text{.}\)
Exercise Group 9.3.10 . Mixed Problems on the Remainder Theorem.
Solve the following problems using the remainder theorem.
(a) \(p(x)=3x^4 + kx^2 + 7\) is divided by \(x-1\text{,}\) the remainder is the same as when \(f(x)=x^4 + kx - 4\) is divided by \(x-2\text{.}\) Find \(k\text{.}\)
(b) \(x^3 + kx^2 - 2x - 7\) is divided by \(x+1\text{,}\) the remainder is \(5\text{.}\) What is the remainder when it is divided by \(x-1\text{?}\)
(c) \(p(x)=x^3 - r x^2 + 3x + r^2\text{,}\) find all values of \(r\) so that \(p(3)=18\text{.}\)
(d) \(x^n + x - 8\) is divided by \(x-2\text{,}\) the remainder is \(10\text{.}\) Find \(n\text{.}\)
(e) \(x^2 - 4x + 3\) is divided by \(x+a\text{,}\) the remainder is \(8\text{.}\) Find all possible values of \(a\text{.}\) Answer .
\(a = 1\) \(a=-5\text{.}\)
