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1.9: Polynomials

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    45151

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    A polynomial is a sum of monomials. So, expressions like:

    \[x^{2}+3 x+7\nonumber\]

    \[-2 x^{3}+4 x^{2}-5 x+2\nonumber\]

    \[x+x^{2}\nonumber\]

    \[-4 x^{3}\nonumber\]

    \[x^{2} y+\dfrac{x y z^{2}}{6}-8 y^{2} z^{2}\nonumber\]

    are examples of polynomials. However, the following are not polynomials:

    \[\dfrac{x^{2}+3 x+4}{x+5}\nonumber\]

    \[\dfrac{2 x^{2} y z^{3}}{-x y^{2}}\nonumber\]

    \[-4 x \sqrt{6 x}\nonumber\]

    The degree of a polynomial is the highest power of the variable(s) that has a non-zero coefficient.

    Example \(\PageIndex{1}\)

    The degree of \(-2 x^{3}+4 x^{2}-5 x+2\) is \(3\).

    Example \(\PageIndex{2}\)

    Determine whether the given expression is a polynomial and if so, find its degree.

    a) \(5 x^{3}+4 x^{2}-3 x+7\)

    This is a polynomial of degree 3.

    b) \(\dfrac{2}{x^{2}+3 x-4}\)

    This is a not a polynomial.

    c) \(2 x^{3}+5 x^{4}+3 x-8\)

    This is a polynomial of degree 4.

    d) \(6.2 \times 10^{-5} x^{8}\)

    This is a polynomial of degree 8.

    A polynomial with one term is called a monomial. For example, \(2 a^{5}\) and \(-3 x y^{2}\) are monomials. A polynomial with two terms is called a binomial. \(5 x^{2}+3 x\) is an example of a binomial. A polynomial with three terms is called a trinomial. \(3 x^{2}+5 x-1\) is a trinomial. It has three terms: \(3 x^{2}, 5 x\) and -1

    Just as we did in chapter 4 when evaluating expressions, we can evaluate polynomials as well.

    Example \(\PageIndex{3}\)

    Evaluate the given polynomial at the given value of the variable(s):

    1. \(3 x+7\) when \(x=2\): \[3(2)+7=6+7=13 \nonumber\]
    2. \(x^{2}+3 x+2\) when \(x=5\): \[(5)^{2}+3(5)+2=25+15+2=42 \nonumber\]
    3. \(2 x^{3}+4 x^{2}-3 x\) when \(x=-2\): \[2(-2)^{3}+4(-2)^{2}-3(-2)=2(-8)+4(4)-(-6)=-16+16+6=6 \nonumber\]
    4. \(-4 x^{7}-3 x^{4}\) when \(x=-1\): \[-4(-1)^{7}-3(-1)^{4}=-4(-1)-3(1)=4-3=1 \nonumber\]
    5. \(2 x^{2} y-5 x^{3} y^{2}\) when \(x=-3\) and \(y=2\):

    \[\begin{align*}
    2(-3)^{2}(2)-5(-3)^{3}(2)^{2} &=2(9)(2)-5(-27)(4) \\
    &=18(2)+135(4) \\
    &=36+540 \\
    &=576
    \end{align*}\]

    Function Notation

    A specific kind of notation, called function notation, can be used to represent polynomials. This notation uses a letter (the name of the function) and the variable at hand (for example, \(x .)\)

    Example \(\PageIndex{4}\)

    1. \(f(x)=3 x+7\) is representing the polynomial \(3 x+7\) as a function called \(f\). The \(x\) in \(f(x)\) is to indicate that the variable in the polynomial is \(x\).
    2. \(g(x)=x^{2}+3 x+2\) is representing the polynomial \(x^{2}+3 x+2\) as a function called \(g \). Again, the \(x\) in \(g(x)\) is to indicate that the variable in the polynomial is \(x\).
    3. \(f(x, y)=2 x^{2} y-5 x^{3} y^{2}\) is representing the polynomial \(2 x^{2} y-5 x^{3} y^{2}\) as a function called \(f \). The \(x\) and \(y\) in \(f(x, y)\) are to indicate that the variables in the polynomial are \(x\) and \(y\).

    You will learn about functions and function notation in a pre-calculus class, but, here we use the notation because it facilitates asking to evaluate a polynomial at a given value of the variable(s), as we saw in Example 7.3.

    So, for example, find \(f(2)\) when \(f(x)=3 x+7\) is asking to evaluate the polynomial \(3 x+7\) when \(x=2 \). So, \(f(2)=3 \cdot 2+7=13\).

    Example \(\PageIndex{5}\)

    \(f(x)=x^{2}-1\). Find \(f(-3)\)

    Solution

    \[f(-3)=(-3)^{2}-1=9-1=8 \nonumber\]

    Example \(\PageIndex{5}\)

    \(g(x)=-3 x^{3}+5 \). Find \(g(-2)\).

    Solution

    \[g(-2)=-3 \cdot(-2)^{3}+5=-3 \cdot(-8)+5=24+5=29 \nonumber\]

    Exit Problem

    Evaluate \(f(-1)\) for the function \(f(x)=-2 x^{3}+3 x^{2}-x\)

    This page titled 1.9: Polynomials is shared under a CC BY-NC-ND 4.0 license and was authored, remixed, and/or curated by Samar ElHitti, Marianna Bonanome, Holly Carley, Thomas Tradler, & Lin Zhou (New York City College of Technology at CUNY Academic Works) .