## 1.5 Polynomial Functions and Complex Zeros Let's dive into the world of polynomial functions, exploring their zeros, multiplicities, and the fascinating interplay between real and complex numbers! 🐰 This section is crucial for understanding the behavior of polynomial functions, a key topic on the AP exam. Expect to see questions that combine concepts from this section with other areas. ## 🔢 Real, Imaginary, and Complex Numbers ### Real Numbers **Real numbers** are all the numbers you can find on a number line. They include: - Integers (e.g., -3, 0, 5) - Fractions (e.g., 1/2, -3/4) - Decimals (e.g., 2.7, -0.5) - Irrational numbers (e.g., √2, π) Real numbers can be positive, negative, or zero. ### Imaginary Numbers **Imaginary numbers** involve the imaginary unit **i**, where $i = \sqrt{-1}$. They are written in the form *bi*, where *b* is a real number. Examples: - 2*i* - -5*i* - (1/3)*i* Imaginary numbers are a subset of complex numbers where the real part is zero. ### Complex Numbers **Complex numbers** combine real and imaginary parts, written in the form **a + bi**, where *a* and *b* are real numbers. Examples: - 3 + 4*i* - -1 - 2*i* - 5 + 0*i* (which is just the real number 5) Complex numbers are the most general form of numbers we're dealing with here. They include both real numbers (when b = 0) and imaginary numbers (when a = 0). ![Diagram of Real, Imaginary, and Complex Numbers](https://brilliant.org/wiki/media/complex-numbers/complex-numbers-venn-diagram.png) *Image Courtesy of [Brilliant](https://brilliant.org/wiki/complex-numbers/)* ## Linear Factors and Multiplicities A **zero** of a polynomial function *p(x)* is a value *a* such that *p(a) = 0*. If *a* is a real number, then **(x - a)** is a linear factor of *p(x)*. 👌 Remember: Zeros are the x-values that make the polynomial equal to zero. They are also called roots. If a polynomial has *real* coefficients, complex zeros always come in **conjugate pairs**. This means if *a + bi* is a zero, then *a - bi* is also a zero. 0️⃣ Real zeros correspond to x-intercepts on the graph of the polynomial. **Multiplicity** refers to how many times a linear factor appears in the factored form of a polynomial. If (x - a) appears *n* times, then *a* is a zero with multiplicity *n*. A polynomial of degree *n* has exactly *n* complex zeros, counting multiplicities. ![Graphs showing different types of roots](https://www.geogebra.org/resource/w7q3y54b/dY2x96R97Z98z42P/material-TMmXTtUQ-thumb@l.png) *Image Courtesy of GeoGebra* ### 🙅🏻 X-Intercepts Real zeros of a polynomial *p(x)* correspond to the **x-intercepts** of its graph. The graph crosses the x-axis at the point **(a, 0)** if *a* is a real zero. 💪 Use x-intercepts to determine intervals where the polynomial is positive or negative. This is a common technique in AP questions. By testing values in intervals between real zeros, you can determine where the function is positive or negative. ↔️ ![Graph showing x and y intercepts](https://www.mathhints.com/wp-content/uploads/2020/01/Polynomial-Example.png) *Image Courtesy of Math Hints* ### ✌🏼 Even Multiplicities If a zero *a* has an **even multiplicity**, the graph of *p(x)* **touches** the x-axis at *x = a* but does not cross it. The function's sign does not change at this zero. ❌ Students often confuse even multiplicities with odd multiplicities. Remember, even multiplicities mean the graph touches the x-axis and turns around. 🤓 ![Graphs of different multiplicities](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_bfe46c_1773.jpg) *Image Courtesy of Lumen Learning* ## Working with Polynomial Functions ### 🧐 Determining Degree of Polynomial Functions The **degree** of a polynomial is the highest power of *x*. You can find it by examining the **successive differences** in the function's output values. ➖ Think of successive differences as layers of change. The layer that becomes constant tells you the degree of the polynomial. Calculate first differences, then second, third, and so on, until you find a constant difference. The number of times you need to take differences will be the degree of the polynomial. ✏️ - Constant first differences: linear (degree 1) - Constant second differences: quadratic (degree 2) - Constant third differences: cubic (degree 3) #### Finding the Degree of Polynomials Practice Question **Example:** Given $p(x) = 2x^3 - 5x^2 + 3x + 1$, let's find the degree using successive differences. 1. First Differences: $p(x+1) - p(x) = 6x^2 - 4x + 0$ 2. Second Differences: $(6(x+1)^2 - 4(x+1)) - (6x^2 - 4x) = 12x + 2$ 3. Third Differences: $(12(x+1) + 2) - (12x + 2) = 12$ Since the third differences are constant, the degree of *p(x)* is 3. 🤩 ### Distinguishing Polynomial Functions: Even Functions **Even functions** are symmetric about the y-axis. They satisfy the property **f(-x) = f(x)**. 🟰 Even functions are like a mirror image across the y-axis. Think of a parabola that opens up or down. Polynomials with only even powers of *x* are even functions. For example: $p(x) = a_n x^n$ where *n* is an even integer. ![Graph of an even function](https://mathbitsnotebook.com/Algebra1/FunctionGraphs/evengraph1.jpg) *Image Courtesy of Math Bits Notebook* ### ⤴️ Distinguishing Polynomial Functions: Odd Functions **Odd functions** are rotationally symmetric about the origin. They satisfy the property **f(-x) = -f(x)**. Odd functions look the same when rotated 180 degrees around the origin. Think of a cubic function. Polynomials with only odd powers of *x* are odd functions. For example: $p(x) = a_n x^n$ where *n* is an odd integer. ![Graph of an odd function](https://mathbitsnotebook.com/Algebra1/FunctionGraphs/oddfunctiongraph.jpg) *Image Courtesy of Math Bits Notebook* ## Final Exam Focus **Key Topics:** - Complex numbers and their properties - Zeros of polynomial functions (real and complex) - Multiplicity of zeros and their impact on the graph - Determining the degree of a polynomial - Even and odd functions **Exam Tips:** - **Time Management:** Don't spend too much time on one question. Move on and come back if time permits. - **Read Carefully:** Pay attention to the wording of the questions, especially when dealing with complex zeros and multiplicities. - **Show Your Work:** Even if you make a mistake, you can get partial credit if you show your reasoning. **Common Pitfalls:** - Forgetting that complex zeros come in conjugate pairs. - Confusing even and odd multiplicities. - Incorrectly applying the method of successive differences. ## Practice Questions **Multiple Choice Questions** 1. What are the zeros of the polynomial $p(x) = (x-2)(x+3)^2$? (A) 2, 3 (B) 2, -3 (C) 2 (multiplicity 1), -3 (multiplicity 2) (D) -2 (multiplicity 1), 3 (multiplicity 2) 2. If a polynomial has a zero of 3 with multiplicity 2, what does this mean for the graph of the polynomial at x=3? (A) The graph crosses the x-axis at x=3 (B) The graph touches the x-axis at x=3 and turns around (C) The graph has a vertical asymptote at x=3 (D) The graph has a hole at x=3 3. Which of the following is an odd function? (A) $f(x) = x^2 + 1$ (B) $f(x) = x^3 + x$ (C) $f(x) = x^4$ (D) $f(x) = x^2 + x$ **Free Response Question** Consider the polynomial function $p(x) = x^4 - 2x^3 - 3x^2 + 8x - 4$. (a) Given that $x = 1$ is a zero of $p(x)$, find all other zeros of $p(x)$, including multiplicities. (3 points) (b) Sketch a graph of $p(x)$, clearly labeling all x-intercepts. (2 points) (c) Determine the intervals where $p(x)$ is positive and negative. (2 points) (d) Is $p(x)$ an even function, an odd function, or neither? Justify your answer. (1 point) **Answer Key and Scoring Rubric for FRQ** (a) **Finding all zeros (3 points):** - Using synthetic division or polynomial long division with (x-1), we get $p(x) = (x-1)(x^3 - x^2 - 4x + 4)$. (1 point) - Factoring by grouping or using synthetic division again, we find that $x=1$ is a root of $x^3 - x^2 - 4x + 4$, so $p(x) = (x-1)(x-1)(x^2-4)$. (1 point) - Factoring the remaining quadratic, we get $p(x) = (x-1)^2(x-2)(x+2)$. Therefore, the zeros are 1 (multiplicity 2), 2 (multiplicity 1), and -2 (multiplicity 1). (1 point) (b) **Sketching the graph (2 points):** - Correctly labeling x-intercepts at -2, 1, and 2. (1 point) - Correctly showing the behavior at x=1 (touching the x-axis) and at x=-2 and x=2 (crossing the x-axis). (1 point) (c) **Intervals of positive and negative (2 points):** - $p(x) > 0$ on the intervals $(-\infty, -2)$, $(2, \infty)$ (1 point) - $p(x) < 0$ on the interval $(-2, 1)$ and $(1, 2)$ (1 point) (d) **Even, odd, or neither (1 point):** - $p(x)$ is neither even nor odd because $p(-x) \neq p(x)$ and $p(-x) \neq -p(x)$. (1 point)

Polynomial Functions and Complex Zeros

Alice White

How are we doing?

Polynomial Functions and Complex Zeros

Alice White

How are we doing?