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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section GR-2 Polynomial Functions

Worksheet Graphing Polynomial Functions

For this outcome Specifically, for this outcome, students should be able to:
  • Given a polynomial in standard form, identify the degree and leading coefficient.
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  • Given a polynomial in factored form, identify the zeros and their multiplicities.
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  • Given a polynomial, identify the y-intercept.
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  • Given a polynomial in standard form, be able to factor the polynomial using synthetic division and the Rational Roots Theorem.
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  • Given a polynomial, identify the end behavior.
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  • Given a polynomial, identify the correct behavior of the function at each zero.
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  • Given a polynomial, be able to sketch the graph of the function.
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Polynomial Functions
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A polynomial Function is a function of the form \(f(x) = c_nx^n + c_{n-1}x^{n-1} + . . . + c_1x + c_0,\text{,}\) where \(c_i\) are constants, \(n\) is a non-negative integer and \(c_n\) is not zero.
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A number \(r\) is said to be a zero of the function \(f\) if \(f(r) = 0\text{.}\)
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If a polynomial is given in it’s factored form \(f(x) = a(x - r_1)^{n_1}(x - r_2)^{n_2} ...(x - r_n)^{n_n}\text{,}\) then the zeros of the function are \(r_1, r_2, ..., r_n\) and the multiplicity of each zero is given by the corresponding exponent \(n_i\text{.}\)
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End Behavior
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The end behavior of a polynomial function is determined by the leading term \(c_nx^n\text{.}\) If \(n\) is even, the ends of the graph go in the same direction (both up or both down). If \(n\) is odd, the ends go in opposite directions (one up and one down). The sign of \(c_n\) determines whether the end as \(x \to \infty\) goes up or down. We denote the end behavior of a polynomial function by writing the behavior of the function, dependent on the behavior of the domain variable. In other words, as \(x \to \infty\text{,}\) \(f(x) \to \infty\) and \(x \to -\infty\text{,}\) \(f(x) \to -\infty\)would describe the end behavior of a polynomial with a positive leading coefficient and odd degree.
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Synthetic Division
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Synthetic division is a method for dividing a polynomial by a linear factor of the form \(x - r\text{.}\) It is a shortcut method that allows us to find the quotient and remainder of the division without having to write out the long division process. The steps for synthetic division are as follows:
  1. Write down the coefficients of the polynomial in descending order of degree. If any terms are missing, use a coefficient of zero.
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  2. Write down the zero \(r\) of the linear factor \(x - r\text{.}\)
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  3. Bring down the leading coefficient to start the synthetic division process.
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  4. Multiply the zero \(r\) by the value just brought down and write it under the next coefficient.
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  5. Add the column and write the result below.
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  6. Repeat steps 4 and 5 until you have processed all coefficients.
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The final row of numbers will give you the coefficients of the quotient polynomial, and the last number will be the remainder. If the remainder is zero, then \(x - r\) is a factor of the original polynomial.
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Rational Roots Theorem and Synthetic Division
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The Rational Roots Theorem states that if a polynomial has integer coefficients, then any rational root of the polynomial must be of the form \(\frac{p}{q}\text{,}\) where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. This theorem helps us to find all possible rational roots of a polynomial, which we can then test using synthetic division.
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Multiplicity of Zeros
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The multiplicity of a zero refers to the number of times that zero appears as a root of the polynomial. If a zero has multiplicity \(k\text{,}\) then the graph of the polynomial will touch the x-axis at that zero and will either bounce off the x-axis (if \(k\) is even) or cross the x-axis (if \(k\) is odd).
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Graphing Polynomial Functions
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When graphing polynomial functions, it is important to consider the following:
  • The end behavior of the function
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  • The zeros of the function and their multiplicities
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  • The y-intercept of the function
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Worksheet GR-2 Polynomial Functions

End Behavior
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Roots and Multiplicities
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Long Division of Polynomials
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Synthetic Division of Polynomials
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The Division Algorithm
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Graphs of Polynomial Functions
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Section GR-2 Rubric

Worksheet Worksheet

Before attempting the GR-2 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 15. GR2 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner No algebraic errors OR some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled. Graph is not labeled and has some of the important features. No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution. Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution. Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
End behavior of polynomials The end-behavior is stated in the correct order using correct notation, and the behavior given is correct. The end-behavior is stated using correct notation, and the behavior given is correct. There is a statement of end-behavior. No evidence of end-behavior was given.
Find zeros and corresponding multiplicities Used the possible rational roots theorem (or factoring methods) to find the factored form of the polynomial; Used this form to state the zeros and their corresponding multiplicities correctly. Used the possible rational roots theorem (or factoring methods) to find the factored form of the polynomial with minor errors; Used this form to state the zeros and their corresponding multiplicities with minor errors. No algebraic method shown to find the factored form of the polynomial; Incorrectly stated the zeros or their multiplicities. No zeros or multiplicities were given.  No evidence showing understanding of the possible rational roots theorem or other factoring techniques.
Find y intercept The correct y-intercept was given using correct notation. The correct y-intercept was given. A point was given that was not the y-intercept. No intercept was given.
Polynomial Graph Graph contains all relevant information; has correct end-behavior, turning points, intercepts, behavior at intercepts, and slopes. Graph contains most of the relevant information; has partially correct end-behavior, slopes, turning points, intercepts or behavior at intercepts. Relevant parts of the graph are missing. errors in the end-behavior, slopes, turning points, intercepts, or behavior at intercepts. No attempt at graphing the function was given.
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Worksheet GR-2 Polynomial Functions

This outcome covers polynomial functions. You should be able to identify the degree and leading coefficient of a polynomial function. Be able to determine the end behavior of a polynomial function. You should also be able to find the x-intercepts and y-intercept of a polynomial function. You should also be able to graph polynomial functions. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
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For the quiz on this outcome you would be given 30 minutes to complete 2 problems. Here are samples of the types of problems you will encounter:
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1.

Given the function \(f(x)=-3(x-1)^3(x+2)(x+1)^2\text{.}\)
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(a)
State the degree and leading coefficient of \(f(x)\text{.}\)
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Hint.
Look at the exponents of each factor to find the degree. Look at the coefficient of the highest degree term to find the leading coefficient.
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Answer.
Degree: 6, Leading Coefficient: -3
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Solution.
To find the degree of the polynomial, we add up the exponents of the factors. The degree is 3 + 1 + 2 = 6.
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(b)
State the end behavior of \(f(x)\text{,}\) using correct notation.
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Hint.
Since the degree is even and the leading coefficient is negative, the end behavior will be the same on both sides and will go down.
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Answer.
As \(x \to \infty\text{,}\) \(f(x) \to -\infty\text{.}\) As \(x \to -\infty\text{,}\) \(f(x) \to -\infty\text{.}\)
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(c)
State the x-intercepts and y-intercept of \(f(x)\text{.}\)
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Hint.
To find the x-intercepts, set each factor equal to zero and solve for x. To find the y-intercept, evaluate the function at x=0.
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Answer.
x-intercepts: (1,0), (-2,0), (-1,0); y-intercept: (0, -6)
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Solution.
To find the x-intercepts, we set each factor equal to zero and solve for x. Setting \((x-1)^3=0\) gives us the x-intercept (1,0). Setting \((x+2)=0\) gives us the x-intercept (-2,0). Setting \((x+1)^2=0\) gives us the x-intercept (-1,0).
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2.

Given the function \(g(x)=2x^4+8x^3+4x^2-8x-6\text{.}\)
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(a)
State the degree and leading coefficient of \(g(x)\text{.}\)
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Hint.
Look at the highest degree term to find the degree and leading coefficient.
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Answer.
Degree: 4, Leading Coefficient: 2
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Solution.
To find the degree of the polynomial, we look at the highest degree term, which is \(2x^4\text{.}\) The degree is 4 and the leading coefficient is 2.
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(b)
State the end behavior of \(g(x)\text{,}\) using correct notation.
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Hint.
Since the degree is even and the leading coefficient is positive, the end behavior will be the same on both sides and will go up.
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Answer.
As \(x \to \infty\text{,}\) \(g(x) \to \infty\text{.}\) As \(x \to -\infty\text{,}\) \(g(x) \to \infty\text{.}\)
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(c)
State the x-intercepts and y-intercept of \(g(x)\text{.}\)
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Hint.
To find the x-intercepts, set the function equal to zero and solve for x. To find the y-intercept, evaluate the function at x=0.
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Answer.
x-intercepts: (1,0), (-1,0), (-3,0); y-intercept: (0, -6)
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Solution.
To find the x-intercepts, we set the function equal to zero and solve for x.We can use the Rational Root Theorem to find possible rational roots, which are factors of the constant term (-6) divided by factors of the leading coefficient (2). Then use synthetic division to test these possible roots until we find one that works (gives a remainder of zero). Setting \(2x^4+8x^3+4x^2-8x-6=0\) gives us the x-intercepts (1,0), (-1,0), and (-3,0). To find the y-intercept, we evaluate the function at x=0: \(g(0)=2(0)^4+8(0)^3+4(0)^2-8(0)-6=-6\text{.}\) Therefore, the y-intercept is (0, -6).
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(d)
Sketch the graph of \(g(x)\text{,}\) showing the correct intercepts and general behavior.
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Hint.
Use the end behavior, intercepts, behavior at the roots and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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3.

Given the function \(p(x)=-x^4-2x^3+3x^2+4x-4\text{.}\)
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(a)
State the end behavior of \(p(x)\text{,}\) using correct notation.
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Hint.
Since the degree is even and the leading coefficient is negative, the end behavior will be the same on both sides and will go down.
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(b)
Write \(p(x)\) in factored form.
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Hint.
Use the Possible Rational Root Theorem to find a root, then use synthetic division to factor the polynomial. Repeat this process until the polynomial is completely factored.
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Answer.
\(p(x)=-1(x-1)(x+2)(x^2+x-2)\)
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Solution.
To factor the polynomial, we first use the Possible Rational Root Theorem to find a root. We test the factors of the constant term (-4) and the leading coefficient (-1). We find that x=1 is a root. We then use synthetic division to divide the polynomial by (x-1). This gives us a quotient of \(-1(x^3+x^2-2x-4)\text{.}\) We then factor the quotient using synthetic division again, finding that x=-2 is a root. This gives us a quotient of \(-1(x^2+x-2)\text{.}\) We can then factor this further to get \(p(x)=-1(x-1)(x+2)(x-1)(x+2)\text{.}\)
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(c)
Sketch the graph of \(p(x)\text{,}\) showing the correct intercepts and general behavior.
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Hint.
Use the end behavior, intercepts, behavior at the roots and general shape of the graph to sketch it.
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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