Polynomials II

Finding Zeros of a Polynomial Function

Zeros of polynomial are the points where the polynomial equals zero on the whole. In simple words, we can say that zeros of polynomial are values of the variable such that the polynomial equals 0 at that point. Zeros of a polynomial are also referred to as the roots of the equation and are often designated as α, β, γ respectively. Some of the methods used to find the zeros of polynomial are grouping, factorization, and using algebraic expressions.

⭐ “Zeros of Polynomial – Formulas, Equations, Examples, Sum And Product”. 2022. CUEMATH. https://www.cuemath.com/algebra/zeros-of-polynomial/.

The Rational Zeros (or Roots) Theorem

If P(x) is a polynomial with integer coefficients and if is p/q a zero of P(x) (i.e., P(p/q) = 0), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial. Here are the steps:

1. Arrange the polynomial in descending order
2. Write down all the factors of the constant term. These are all the possible values of p.
3. Write down all the factors of the leading coefficient. These are all the possible values of q.
4. Write down all the possible values of p/q. Remember that since factors can be negative, p/q and –p/q must both be included. Simplify each value and cross out any duplicates.
5. Use synthetic division to determine the values of p/q for which P(p/q) = 0. These are all the rational roots of P(x).

Example: Find all the rational zeros of P(x) = x³ – 9x + 9 + 2x⁴ – 19x².

1. P(x) = 2x⁴ + x³ -19x² – 9x + 9
2. Factors of constant term: ±1, ±3, ±9.
3. Factors of leading coefficient: ±1, ±2.
4. Possible values of : ±, ±, ±, ±, ±, ±. These can be simplified to: ±1, ±, ±3, ±, ±9, ±.
5. Use synthetic division:

Thus, the rational roots of P(x) are x = – 3, -1, 1/2, and 3.

We can often use the rational zeros theorem to factor a polynomial. Using synthetic division, we can find one real root a and we can find the quotient when P(x) is divided by x – a. Next, we can use synthetic division to find one factor of the quotient. We can continue this process until the polynomial has been completely factored.

References

“Algebra – Finding Zeroes Of Polynomials “. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/classes/alg/findingzeroesofpolynomials.aspx.

“Algebra II: Polynomials: The Rational Zeros Theorem | sparknotes”. 2022. sparknotes. https://www.sparknotes.com/math/algebra2/polynomials/section4/.

Estela, Mike. 2022. “Rational Roots Test – Chilimath”. Chilimath. https://www.chilimath.com/lessons/intermediate-algebra/rational-roots-test/.

“Finding Zeros Of A Polynomial Function | College Algebra”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/waymakercollegealgebra/chapter/zeros-of-a-polynomial-function/.

“Rational Root Theorem (Rational Zero Theorem) – Examples, Proof”. 2022. CUEMATH. https://www.cuemath.com/algebra/rational-root-theorem/.

“Rational Zero Theorem”. 2022. CliffsNotes. https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/polynomial-functions/rational-zero-theorem.

Hrankowski, Adam. “The Rational Root Theorem”. 2020. Medium. https://medium.com/mathadam/the-rational-root-theorem-62df4d43329c.

 

⭐ “Zeroes and Their Multiplicities”. 2022. purplemath.com. https://www.purplemath.com/modules/polyends2.htm.

The Fundamental Theorem of Algebra

A polynomial p(x) of degree n ≥ 1 with complex coefficients has, counted with multiplicity, exactly n roots.

The part “counted with multiplicity” means that we have to count the roots by their multiplicity, that is, by the times they are repeated. For example, in the equation (x-2)³(x+2) = 0, we have a polynomial of degree four.

However, we can only count two real roots. This is because the root at x = 2 is a multiple root with a multiplicity of three. Therefore, the total number of roots, when counting multiplicity, is four.

The Fundamental Theorem of Algebra stated another way:

If P(x) is a polynomial of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots.

Read carefully to fully understand what this theorem is saying.

• “including multiple… roots” – if a polynomial has a repeated root, each repetition of the root is counted.
• “including… complex roots” – the term “complex” is referring to “complex roots with a non-zero imaginary part” (a + bi with b ≠ 0) which implies the conjugate of the complex root will also be counted since such complex roots come in conjugate pairs.

References

“Complex Roots Of A Polynomial – Examples And Practice Problems – MECHAMATH”. 2021. MECHAMATH. https://www.mechamath.com/algebra/complex-roots-of-a-polynomial/#2-how-to-know-how-many-complex-roots-a-polynomial-has.

“Fundamental Theorem Of Algebra”. 2022. mathsisfun.com. https://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html.

Roberts, Donna. 2022. “Fundamental Theorem Of Algebra – MathBitsNotebook (A2 – CCSS Math)”. mathbitsnotebook.com. https://mathbitsnotebook.com/Algebra2/Polynomials/POfundamentalThm.html.

Interpreting Turning Points

A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). A polynomial of degree n will have at most n – 1 turning points.

Example: Finding the Maximum possible Number of Turning Points Using the Degree of a Polynomial Function

Find the maximum possible number of turning points of each polynomial function.

Solution

a. f(x) = -x³ + 4x⁵ – 3x² + 1

First, rewrite the polynomial function in descending order: 4x⁵ – x³ – 3x² + 1

Identify the degree of the function. This polynomial function is of degree 5.

The maximum possible number of turning points is 5 – 1 = 4.

b. f(x) = -(x – 1)²(1 + 2x²)

First, identify the leading term of the polynomial function if the function were expanded.

-2x⁴ + 4x³ -3x² + 2x – 1

Then, identify the degree of the polynomial function. This polynomial function is of degree 4.
The maximum possible number of turning points is 4 – 1 = 3.

References

“3.4: Graphs Of Polynomial Functions”. 2019. Mathematics LibreTexts. https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_206_Precalculus/3%3A_Polynomial_and_Rational_Functions_New/3.4%3A_Graphs_of_Polynomial_Functions.

Roberts, Donna. 2022. “More Features Of Function Graphs Refresher – MathBitsNotebook (A2 – CCSS Math)”. mathbitsnotebook.com. https://mathbitsnotebook.com/Algebra2/Polynomials/POFunctionFeaturesMore.html.

Graphing a Polynomial Function

Graphing a polynomial function involves the following steps.

1. Factor the polynomial (if needed)
2. Determine the End Behavior by examining the leading term (see the Leading Coefficient Test).
3. Find the intercepts and use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts.
4. Use the end behavior and the behavior at the intercepts to sketch a graph.
5. Optionally …
   • Check for symmetry. If the function is an even function, its graph is symmetrical about the y-axis, that is, f(−x) = f(x) . If a function is an odd function, its graph is symmetrical about the origin, that is, f(−x) = −f(x).
   • Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
   • Use technology (e.g., Desmos or a graphing calculator) to check the graph.

Example

• Factor the polynomial 2x⁴ – 3x³ – 21x² – 2x + 24
  • (2x + 3)(x – 1)(x + 2)(x – 4)
• Determine the end behavior using the Leading Coefficient Test (see Table View of Leading Coefficient Test below)
  • End Behavior: f(x) → +∞, as x → -∞ and f(x) → +∞, as x → +∞
  • Leading Coefficient Test: Since the leading coefficient 2x⁴, because n is even (4) and aⁿ is positive (2), the graph rises to the left and right
• Find the intercepts and use the multiplicities of the zeros to determine the behavior of the polynomial at the x-intercepts.
  • Intercepts: (-3/2, 0), (1, 0), (-2, 0), (4, 0)
  • Multiplicities: Each factor has a multiplicity of 1, therefore, the graph passes through the x-axis at all x-intercepts
• Use the end behavior and the behavior at the intercepts to sketch a graph

Leading Coefficient Test

All polynomial functions of first or higher order either increase or decrease indefinitely as x values grow larger and smaller. It is possible to determine the end behavior (i.e., the behavior when tends to infinity) of a polynomial function without using a graph. Consider the polynomial function:

a_nxⁿ is called the leading term of f(x), while a_n ≠ 0 is known as the leading coefficient. The properties of the leading term and leading coefficient indicate whether f(x) increases or decreases continually as the x-values approach positive and negative infinity:

• If n is odd and a_n is positive, the function declines to the left and inclines to the right.
• If n is odd and a_n is negative, the function inclines to the left and declines to the right.
• If n is even and a_n is positive, the function inclines both to the left and to the right.
• If n is even and a_n is negative, the function declines both to the left and to the right.

The leading coefficient of a polynomial can change how steep the graph of a polynomial is.

References

⭐ “Leading Coefficient Test”. 2022. varsitytutors.com. https://www.varsitytutors.com/hotmath/hotmath_help/topics/leading-coefficient-test.

End Behavior

If f is a function whose domain and range are subsets of real numbers, the end behavior of a function is what happens to the values of the function as the values of x approach positive infinity and negative infinity.

The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity.

The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.

The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. So, the sign of the leading coefficient is sufficient to predict the end behavior of the function.

References

“End Behavior Of A Function”. 2022. varsitytutors.com. https://www.varsitytutors.com/hotmath/hotmath_help/topics/end-behavior-of-a-function.

Multiplicity and Turning Points

If a polynomial contains a factor of the form (x−h)^p , the behavior near the x-intercept is determined by the power p . We say that x = h is a zero of multiplicity p.

• The graph of a polynomial function will touch the x-axis at zeros with even multiplicities.
• The graph will cross the x-axis at zeros with odd multiplicities.
• The higher the multiplicity, the flatter the curve is at the zero.
• The sum of the multiplicities is the degree of the polynomial function.

For zeros with even multiplicities, the graphs touch or are tangent to the x-axis. For zeros with odd multiplicities, the graphs cross
or intersect the x-axis. See the figure below for examples of graphs of polynomial functions with a zero of multiplicity 1 (y = x), 2 (y = x²), and 3 (y = x³).

The graphs clearly show that the higher the multiplicity, the flatter the graph is at the zero.

Turning Points: A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

A polynomial of degree n will have, at most, n x-intercepts and n−1 turning points.

References

⭐ “3.4: Graphs Of Polynomial Functions”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/03%3A_Polynomial_and_Rational_Functions/3.04%3A_Graphs_of_Polynomial_Functions.

“Multiplicity And Turning Points”. 2016. Geogebra. https://www.geogebra.org/m/rzH422H2.

“Multiplicity and Turning Points”. 2022. coursehero.com. https://www.coursehero.com/study-guides/ivytech-wmopen-collegealgebra/multiplicity-and-turning-points/.

Stapel, Elizabeth. 2022. “Polynomial Graphs: Zeroes And Their Multiplicities | Purplemath”. Purplemath. https://www.purplemath.com/modules/polyends2.htm.

Videos

 

This precalculus video tutorial explains how to graph polynomial functions by identifying the end behavior of the function as well as the multiplicity of each zero or x intercept.

 

 

In this video, we explore identifying zeros and their multiplicity of a polynomial in factored form. Also, we observe what effect multiplicities have on the graph of a function.

Writing an Equation from a Polynomial Graph

Graphing a polynomial function involves the following steps.

• Identify the x-intercepts of the graph to find the factors of the polynomial.
• Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.
• Find the polynomial of least degree containing all the factors found in the previous step.
• Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.

STRETCH FACTOR: If a polynomial of lowest degree p has horizontal intercepts at x = x₁, x₂, …, x_n, then the polynomial can be written in the factored form: f(x) = a(x − x₁)^p1(x − x^₂)^p2 ⋯ (x − x_n)^pn where the powers p_i on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor aa can be determined given a value of the function other than the x-intercept.

Example

Write the equation for the polynomial graph shown.

 

STEPS

1. Identify the x-intercepts of the graph: -7, -3, 4, 6, 8
2. Examine the behavior of the graph at the x-intercepts
   • At x = -7, the graph goes though the x-axis, therefore, the multiplicity is 1
   • At x = -3, the graph acts like a quadratic, therefore, the multiplicity is 2
   • At x = 4, the graph acts like a cubic, therefore, the multiplicity is 3
   • At x = 6, the graph goes though the x-axis, therefore, the multiplicity is 1
   • At x = 8, the graph goes though the x-axis, therefore, the multiplicity is 1
3. Find the polynomial of least degree containing all the factors found in the previous step.
   • y = (x + 7)(x + 3)²(x – 4)³(x – 6)(x – 8)
4. Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.
   • Use the point (0, 3) to determine the stretch factor
   • f(0) = a(0 + 7)(0 + 3)²(0 – 4)³(0 – 6)(0 – 8) = a * 7 * 9 * -64 * -6 * -8 = -193536a
   • 3 = -193536a
   • a = -3/193536 (-0.00001550099)
   • The graphed polynomial appears to represent the function
     f(x) = (-3/193536)(x + 7)(x + 3)²(x – 4)³(x – 6)(x – 8),
     or -(x + 7)(x + 3)²(x – 4)³(x – 6)(x – 8) (What the teacher is probably looking for.)

References

⭐ “3.4: Graphs Of Polynomial Functions”. 2020. Mathematics LibreTexts. https://math.libretexts.org/Courses/Monroe_Community_College/MTH_165_College_Algebra_MTH_175_Precalculus/03%3A_Polynomial_and_Rational_Functions/3.04%3A_Graphs_of_Polynomial_Functions.

Videos

 

Finding the equation of a polynomial from a graph by writing out the factors. This example has a double root. I show you how to find the factors and the leading coefficient.

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The featured image on this page is from the YouTube video “What is a Polynomial?” 2012. ExamSolutions. https://youtu.be/lYy8Nh9AdWk.

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⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.