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Wize AP Calculus (BC) Textbook > Review: Functions/Pre-Calculus

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?
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Polynomial Graphs & Characteristics

A polynomial function of degree 'nn' can be expressed as:
f(x)=axn+bxn−1+ ... +cx+d\displaystyle \boxed{f(x) =ax^n+bx^{n-1}+~...~+cx+d} , where a,b,c,d∈Ra, b, c, d\in\mathbb{R}.
  • The degree of the polynomial is 'nn'.
  • The leading coefficient is 'aa'.
  • A function is increasing xxwhenever & yy increase and decrease simultaneously.
  • A function is decreasing whenever xx is increasing & yy is decreasing (or vice versa).

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Positive, Odd Polynomials

When a>0  &  n is odda>0~~\&~~n~\text{is odd}:


f(x)=xf(x)=x3f(x)=x5Degree135SymmetryPointPointPointMin # of111X-IntMax # of135X-IntEnd Behaviourx→∞, y→∞x→∞, y→∞x→∞, y→∞x→−∞, y→−∞x→−∞, y→−∞x→−∞, y→−∞Increasing(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)DecreasingNANANA\begin{array}{|c|c|c|c|}\hline\\ &f(x)=x&f(x)=x^3&f(x)=x^5\\\\\hline\\ \textbf{Degree}&1&3&5\\\\\hline\\ \textbf{Symmetry}&\text{Point}&\text{Point}&\text{Point}\\\\\hline\\ \textbf{Min \# of}&1&1&1\\\textbf{X-Int}&\\\\\hline\\ \textbf{Max \# of}&1&3&5\\\textbf{X-Int}&\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)\\\\\hline\\ \textbf{Decreasing}&\text{NA}&\text{NA}&\text{NA}\\\\\hline \end{array}

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Negative, Odd Polynomials

When a<0  &  n is odda<0~~\&~~n~\text{is odd}:

f(x)=−xf(x)=−x3f(x)=−x5Degree135SymmetryPointPointPointMin # of111X-IntMax # of135X-IntEnd Behaviourx→∞, y→−∞x→∞, y→−∞x→∞, y→−∞x→−∞, y→∞x→−∞, y→∞x→−∞, y→∞IncreasingNANANADecreasing(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)\begin{array}{|c|c|c|c|}\hline\\ &f(x)=-x&f(x)=-x^3&f(x)=-x^5\\\\\hline\\ \textbf{Degree}&1&3&5\\\\\hline\\ \textbf{Symmetry}&\text{Point}&\text{Point}&\text{Point}\\\\\hline\\ \textbf{Min \# of}&1&1&1\\\textbf{X-Int}&\\\\\hline\\ \textbf{Max \# of}&1&3&5\\\textbf{X-Int}&\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin\\&x\rightarrow-\infin,~y\rightarrow\infin&x\rightarrow-\infin,~y\rightarrow\infin&x\rightarrow-\infin,~y\rightarrow\infin\\\\\hline\\ \textbf{Increasing}&\text{NA}&\text{NA}&\text{NA}\\\\\hline\\ \textbf{Decreasing}&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)&(-\infin,0)\cup(0,\infin)\\\\\hline \end{array}

Wize Concept
For odd polynomial functions, the minimum number of real roots is 1 and the maximum number of real roots is 'nn'.

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Positive, Even Polynomials

When a>0  &  n is evena>0~~\&~~n~\text{is even}:
f(x)=x2f(x)=x4Degree24SymmetryLineLineMin # of02X-IntMax # of04X-IntEnd Behaviourx→∞, y→∞x→∞, y→∞x→−∞, y→−∞x→−∞, y→−∞Increasing(0,∞)(0,∞)Decreasing(−∞,0)(−∞,0)\begin{array}{|c|c|c|}\hline\\ &f(x)=x^2&f(x)=x^4\\\\\hline\\ \textbf{Degree}&2&4\\\\\hline\\ \textbf{Symmetry}&\text{Line}&\text{Line}\\\\\hline\\ \textbf{Min \# of}&0&2\\\textbf{X-Int}\\\\\hline\\ \textbf{Max \# of}&0&4\\\textbf{X-Int}\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow\infin&x\rightarrow\infin,~y\rightarrow\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(0,\infin)&(0,\infin)\\\\\hline\\ \textbf{Decreasing}&(-\infin,0)&(-\infty,0)\\\\\hline \end{array}
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Negative, Even Polynomials

When a<0  &  n is evena<0~~\&~~n~\text{is even}:
f(x)=−x2f(x)=−x4Degree24SymmetryLineLineMin # of02X-IntMax # of04X-IntEnd Behaviourx→∞, y→−∞x→∞, y→−∞x→−∞, y→−∞x→−∞, y→−∞Increasing(−∞,0)(−∞,0)Decreasing(0,∞)(0,∞)\begin{array}{|c|c|c|}\hline\\ &f(x)=-x^2&f(x)=-x^4\\\\\hline\\ \textbf{Degree}&2&4\\\\\hline\\ \textbf{Symmetry}&\text{Line}&\text{Line}\\\\\hline\\ \textbf{Min \# of}&0&2\\\textbf{X-Int}\\\\\hline\\ \textbf{Max \# of}&0&4\\\textbf{X-Int}\\\\\hline\\ \textbf{End Behaviour}&x\rightarrow\infin,~y\rightarrow-\infin&x\rightarrow\infin,~y\rightarrow-\infin\\&x\rightarrow-\infin,~y\rightarrow-\infin&x\rightarrow-\infin,~y\rightarrow-\infin\\\\\hline\\ \textbf{Increasing}&(-\infin,0)&(-\infin,0)\\\\\hline\\ \textbf{Decreasing}&(0,\infin)&(0,\infin)\\\\\hline \end{array}

Wize Concept
For even polynomial functions, the minimum number of real roots is 0 and the maximum number of real roots is 'nn'.

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Example: Polynomial Graphs & Characteristics

Based on the following graph:


Find the following:
  • Number of xx-intercepts
  • End Behaviour
  • Odd or Even Degree
  • Positive or Negative Leading Coefficient
  • Degree of Polynomial
Number of xx-intercepts: 4

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

Odd or Even Degree: Even

Positive or Negative Leading Coefficient: Positive

Degree of Polynomial: 4
Based on the graph below, answer the following questions.