Polynomial Functions and Complex Zeros

#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)

#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:

• 2i
• -5i
• (1/3)i

#Complex Numbers

Complex numbers combine real and imaginary parts, written in the form a + bi, where a and b are real numbers.

Examples:

• 3 + 4i
• -1 - 2i
• 5 + 0i (which is just the real number 5)

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#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). 👌

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

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.

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#🙅🏻 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. 💪

By testing values in intervals between real zeros, you can determine where the function is positive or negative. ↔️

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#✌🏼 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. ❌

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#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. ➖

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. ✏️

#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). 🟰

Polynomials with only even powers of x are even functions. For example: $p(x) = a_n x^n$ where n is an even integer.

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).

Polynomials with only odd powers of x are odd functions. For example: $p(x) = a_n x^n$ where n is an odd integer.

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

#Practice Questions

Practice Question

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)