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Wize High School Grade 12 Pre-Calculus Textbook > Polynomial Functions

Polynomials

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Basic of Polynomials

A polynomial is an expression consisting of one or more algebraic terms with coefficients, constants, & variables raised to whole-number exponents.

The degree of a polynomial is determined by the value of the greatest exponent.

The following are examples of polynomials and their degree:
2x2+5xDegree 22a−15a3Degree 3−6y6−4y2Degree 6\begin{array}{rcccl} 2x^2+5x&&&\text{Degree 2}\\\\ \sqrt{2}a-\frac{1}{5}a^3&&&\text{Degree 3}\\\\ -6y^6-4y^2&&&\text{Degree 6} \end{array}

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Types of Polynomials

  • A monomial is a polynomial consisting of one term
  • Example: x2, 4a, 3xyzx^2,~4a,~3xyz
  • A binomial is a polynomial consisting of two terms
  • Example: 2x−y, 3ab+12x-y,~3a^b+1
  • A trinomial is a polynomial consisting of three terms
  • Example: ax2+bx+cax^2+bx+c


First Degree Polynomials

  • y=xy=x
  • Linear Monomial
SymmetryPointEven/OddOddX-Intercepts(0,0)Y-Intercepts(0,0)End Behavoiurx → ∞, y →∞x → −∞, y →−∞Min/MaxNoneDomain(−∞,∞)Range(−∞,∞)\begin{array}{|c|c|}\hline\\ \textbf{Symmetry}&\text{Point}\\\\\hline\\ \textbf{Even/Odd}&\text{Odd}\\\\\hline\\ \textbf{X-Intercepts}&(0,0)\\\\\hline\\ \textbf{Y-Intercepts}&(0,0)\\\\\hline\\ \textbf{End Behavoiur}&x~\rightarrow~\infin,~y~\rightarrow\infin\\ &x~\rightarrow~-\infin,~y~\rightarrow-\infin\\\\\hline\\ \textbf{Min/Max}&\text{None}\\\\\hline\\ \textbf{Domain}&(-\infin,\infin)\\\\\hline\\ \textbf{Range}&(-\infin,\infin)\\\\\hline \end{array}
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Second Degree Polynomials

  • y=x2y=x^2
  • Quadratic Monomial
SymmetryLineEven/OddEvenX-Intercepts(0,0)Y-Intercepts(0,0)End Behavoiurx → ∞, y →∞x → −∞, y →∞Min/MaxMin at (0,0)Domain(−∞,∞)Range[0,∞)\begin{array}{|c|c|}\hline\\ \textbf{Symmetry}&\text{Line}\\\\\hline\\ \textbf{Even/Odd}&\text{Even}\\\\\hline\\ \textbf{X-Intercepts}&(0,0)\\\\\hline\\ \textbf{Y-Intercepts}&(0,0)\\\\\hline\\ \textbf{End Behavoiur}&x~\rightarrow~\infin,~y~\rightarrow\infin\\ &x~\rightarrow~-\infin,~y~\rightarrow\infin\\\\\hline\\ \textbf{Min/Max}&\text{Min at}~(0,0)\\\\\hline\\ \textbf{Domain}&(-\infin,\infin)\\\\\hline\\ \textbf{Range}&[0,\infin)\\\\\hline \end{array}
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Third Degree Polynomials

  • y=x3y=x^3
  • Cubic Monomial
SymmetryPointEven/OddOddX-Intercepts(0,0)Y-Intercepts(0,0)End Behavoiurx → ∞, y →∞x → −∞, y →−∞Min/MaxNoneDomain(−∞,∞)Range(−∞,∞)\begin{array}{|c|c|}\hline\\ \textbf{Symmetry}&\text{Point}\\\\\hline\\ \textbf{Even/Odd}&\text{Odd}\\\\\hline\\ \textbf{X-Intercepts}&(0,0)\\\\\hline\\ \textbf{Y-Intercepts}&(0,0)\\\\\hline\\ \textbf{End Behavoiur}&x~\rightarrow~\infin,~y~\rightarrow\infin\\ &x~\rightarrow~-\infin,~y~\rightarrow-\infin\\\\\hline\\ \textbf{Min/Max}&\text{None}\\\\\hline\\ \textbf{Domain}&(-\infin,\infin)\\\\\hline\\ \textbf{Range}&(-\infin,\infin)\\\\\hline \end{array}
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Fourth Degree Polynomials

  • y=x4y=x^4
  • Quartic Monomial
SymmetryLineEven/OddEvenX-Intercepts(0,0)Y-Intercepts(0,0)End Behavoiurx → ∞, y →∞x → −∞, y →∞Min/MaxMin at (0,0)Domain(−∞,∞)Range[0,∞)\begin{array}{|c|c|}\hline\\ \textbf{Symmetry}&\text{Line}\\\\\hline\\ \textbf{Even/Odd}&\text{Even}\\\\\hline\\ \textbf{X-Intercepts}&(0,0)\\\\\hline\\ \textbf{Y-Intercepts}&(0,0)\\\\\hline\\ \textbf{End Behavoiur}&x~\rightarrow~\infin,~y~\rightarrow\infin\\ &x~\rightarrow~-\infin,~y~\rightarrow\infin\\\\\hline\\ \textbf{Min/Max}&\text{Min at}~(0,0)\\\\\hline\\ \textbf{Domain}&(-\infin,\infin)\\\\\hline\\ \textbf{Range}&[0,\infin)\\\\\hline \end{array}
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Fifth Degree Polynomial

  • y=x5y=x^5
  • Quintic Monomial
SymmetryPointEven/OddOddX-Intercepts(0,0)Y-Intercepts(0,0)End Behavoiurx → ∞, y →∞x → −∞, y →−∞Min/MaxNoneDomain(−∞,∞)Range(−∞,∞)\begin{array}{|c|c|}\hline\\ \textbf{Symmetry}&\text{Point}\\\\\hline\\ \textbf{Even/Odd}&\text{Odd}\\\\\hline\\ \textbf{X-Intercepts}&(0,0)\\\\\hline\\ \textbf{Y-Intercepts}&(0,0)\\\\\hline\\ \textbf{End Behavoiur}&x~\rightarrow~\infin,~y~\rightarrow\infin\\ &x~\rightarrow~-\infin,~y~\rightarrow-\infin\\\\\hline\\ \textbf{Min/Max}&\text{None}\\\\\hline\\ \textbf{Domain}&(-\infin,\infin)\\\\\hline\\ \textbf{Range}&(-\infin,\infin)\\\\\hline \end{array}
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Example: Basics of Polynomials

Let f(x)=x(x−1)(x−2) f(x)=x(x-1)(x-2)~ be shown below:


Identify the following:
  • Degree
  • End Behaviour
  • X-Intercepts
  • Domain & Range
Degree: 3

End Behaviour:
x→∞, y→∞x\rightarrow\infty,~y\rightarrow\infty
x→−∞, y→−∞x\rightarrow-\infin,~y\rightarrow-\infin

xx-Intercepts: x = 0, 1, 2

Domain: (−∞,∞)(-\infin,\infin)

Range: (−∞,∞)(-\infin,\infin)
Which of the following are polynomials?