Essential Concepts
Dividing Polynomials
Polynomial long division works similarly to long division with numbers. We divide the leading terms, multiply, subtract, and repeat until we reach the end of the dividend.
The process:
- Set up the division problem
- Divide the leading term of the dividend by the leading term of the divisor
- Multiply the answer by the divisor and write below like terms
- Subtract the bottom expression from the terms above
- Bring down the next term of the dividend
- Repeat steps 2-5 until reaching the last term
- Express any non-zero remainder as a fraction with the divisor as denominator
The Division Algorithm
For polynomial dividend [latex]f(x)[/latex] and non-zero divisor [latex]d(x)[/latex]:
[latex]f(x) = d(x)q(x) + r(x)[/latex]
where:
- [latex]q(x)[/latex] is the quotient
- [latex]r(x)[/latex] is the remainder (degree less than [latex]d(x)[/latex] or equals zero)
- If [latex]r(x) = 0[/latex], then [latex]d(x)[/latex] divides evenly into [latex]f(x)[/latex]
Synthetic Division
Synthetic division is a shortcut for dividing polynomials by binomials of the form [latex]x - k[/latex].
Steps:
- Write [latex]k[/latex] for the divisor (if divisor is [latex]x + 2[/latex], then [latex]k = -2[/latex])
- Write coefficients of the dividend (use 0 for missing terms)
- Bring down the leading coefficient
- Multiply by [latex]k[/latex] and write the product in the next column
- Add the terms in that column
- Repeat multiply-and-add process for remaining columns
- Bottom row gives quotient coefficients; last number is the remainder
Complex Numbers
Imaginary Numbers
The imaginary unit [latex]i[/latex] is defined as [latex]i = \sqrt{-1}[/latex], which means [latex]i^2 = -1[/latex].
Any square root of a negative number can be written as a multiple of [latex]i[/latex]: [latex]\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i[/latex]
Complex Numbers
A complex number has the form [latex]a + bi[/latex] where:
- [latex]a[/latex] is the real part
- [latex]b[/latex] is the imaginary part (coefficient of [latex]i[/latex])
- Both [latex]a[/latex] and [latex]b[/latex] are real numbers
The Complex Plane
Complex numbers are plotted on a coordinate system where:
- Horizontal axis = real axis
- Vertical axis = imaginary axis
- The complex number [latex]a + bi[/latex] is plotted at point [latex](a, b)[/latex]
Arithmetic with Complex Numbers
Addition and Subtraction:
Combine real parts and imaginary parts separately
- [latex](a + bi) + (c + di) = (a + c) + (b + d)i[/latex]
- [latex](a + bi) - (c + di) = (a - c) + (b - d)i[/latex]
Multiplication:
Use the distributive property (FOIL), remembering that [latex]i^2 = -1[/latex]
- [latex](a + bi)(c + di) = (ac - bd) + (ad + bc)i[/latex]
Multiplying by a real number:
Distribute to both parts
- [latex]k(a + bi) = ka + kbi[/latex]
Complex Conjugates
The complex conjugate of [latex]a + bi[/latex] is [latex]a - bi[/latex].
When you multiply complex conjugates, the result is always a real number:
- [latex](a + bi)(a - bi) = a^2 + b^2[/latex]
Powers of [latex]i[/latex]
The powers of [latex]i[/latex] cycle in a pattern of 4:
- [latex]i^1 = i[/latex]
- [latex]i^2 = -1[/latex]
- [latex]i^3 = -i[/latex]
- [latex]i^4 = 1[/latex]
- [latex]i^5 = i[/latex] (pattern repeats)
Zeros of Polynomial Functions
Remainder Theorem
If a polynomial [latex]f(x)[/latex] is divided by [latex]x - k[/latex], then the remainder is [latex]f(k)[/latex].
Factor Theorem
[latex]k[/latex] is a zero of [latex]f(x)[/latex] if and only if [latex](x - k)[/latex] is a factor of [latex]f(x)[/latex].
This means:
- If [latex]f(k) = 0[/latex], then [latex](x - k)[/latex] is a factor
- If [latex](x - k)[/latex] is a factor, then [latex]f(k) = 0[/latex]
Rational Zero Theorem
For polynomial [latex]f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0[/latex] with integer coefficients:
Every possible rational zero has the form [latex]\frac{p}{q}[/latex] where:
- [latex]p[/latex] is a factor of the constant term [latex]a_0[/latex]
- [latex]q[/latex] is a factor of the leading coefficient [latex]a_n[/latex]
Finding Zeros of Polynomial Functions
Steps:
- Use the Rational Zero Theorem to list all possible rational zeros
- Use synthetic division to test each candidate
- If remainder is 0, the candidate is a zero
- If remainder is not 0, try the next candidate
- Continue until the quotient is quadratic
- Find remaining zeros using factoring or the quadratic formula
Fundamental Theorem of Algebra
Every polynomial function of degree [latex]n > 0[/latex] has at least one complex zero.
This means:
- A polynomial of degree [latex]n[/latex] has exactly [latex]n[/latex] zeros (counting multiplicities)
- These zeros may be real or complex
- Real numbers are complex numbers with imaginary part equal to 0
Linear Factorization Theorem
A polynomial function of degree [latex]n[/latex] can be written as a product of [latex]n[/latex] linear factors:
[latex]f(x) = a(x - c_1)(x - c_2) \cdots (x - c_n)[/latex]
where [latex]c_1, c_2, \ldots, c_n[/latex] are complex numbers (possibly including real numbers).
Complex Conjugate Theorem
If a polynomial function [latex]f[/latex] has real coefficients and [latex]a + bi[/latex] (where [latex]b \ne 0[/latex]) is a zero, then the complex conjugate [latex]a - bi[/latex] must also be a zero.
This means complex zeros always come in conjugate pairs for polynomials with real coefficients.
Writing Polynomials from Zeros
Given zeros and a point on the graph:
- Use the zeros to write linear factors [latex](x - c_1)(x - c_2) \cdots[/latex]
- If a complex zero is given, include its conjugate
- Multiply the factors together: [latex]f(x) = a(\text{factors})[/latex]
- Use the given point to solve for the leading coefficient [latex]a[/latex]
Polynomial Inequalities and Inverses
Solving Polynomial Inequalities
To solve inequalities like [latex]f(x) > 0[/latex] or [latex]f(x) < 0[/latex]:
Method 1 (Test Values):
- Solve the equality [latex]f(x) = 0[/latex] to find zeros
- Zeros divide the number line into intervals
- Choose a test value in each interval
- Evaluate [latex]f(\text{test value})[/latex] to determine if positive or negative
- Select intervals that satisfy the inequality
Method 2 (Graphing):
- Find zeros of the function
- Sketch the graph using end behavior and multiplicity
- Identify where the graph is above ([latex]> 0[/latex]) or below ([latex]< 0[/latex]) the x-axis
Inverse of Polynomial Functions
Not all polynomial functions have inverses that are functions. A polynomial must be one-to-one (pass the horizontal line test) to have an inverse function.
Notation: [latex]f^{-1}(x)[/latex] denotes the inverse function, NOT [latex]\frac{1}{f(x)}[/latex]
Finding Inverses of One-to-One Polynomials
Steps:
- Verify the function is one-to-one
- Replace [latex]f(x)[/latex] with [latex]y[/latex]
- Interchange [latex]x[/latex] and [latex]y[/latex]
- Solve for [latex]y[/latex]
- Rename as [latex]f^{-1}(x)[/latex]
Key properties:
- [latex]f^{-1}(f(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]
- [latex]f(f^{-1}(x)) = x[/latex] for all [latex]x[/latex] in the domain of [latex]f^{-1}[/latex]
- Graphs of [latex]f[/latex] and [latex]f^{-1}[/latex] are reflections across the line [latex]y = x[/latex]
- Domain of [latex]f[/latex] = Range of [latex]f^{-1}[/latex]
- Range of [latex]f[/latex] = Domain of [latex]f^{-1}[/latex]
Restricting the Domain
When a polynomial is not one-to-one (like quadratic functions), we can restrict its domain to make it one-to-one, then find the inverse on that restricted domain.
For [latex]f(x) = (x - h)^2 + k[/latex]:
- Restrict to [latex]x \ge h[/latex] (right side): [latex]f^{-1}(x) = h + \sqrt{x - k}[/latex]
- Restrict to [latex]x \le h[/latex] (left side): [latex]f^{-1}(x) = h - \sqrt{x - k}[/latex]
The outputs of the inverse must match the restricted domain of the original function.
Key Equations
| Imaginary unit | [latex]i = \sqrt{-1}[/latex], therefore [latex]i^2 = -1[/latex] |
| Complex number standard form | [latex]a + bi[/latex] |
Glossary
complex conjugate
the complex number in which the sign of the imaginary part is changed and the real part is left unchanged; when multiplied by or added to the original complex number, the result is a real number
complex number
a number of the form [latex]a + bi[/latex] where [latex]a[/latex] is the real part and [latex]bi[/latex] is the imaginary part
complex plane
a coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number
Descartes’ Rule of Signs
a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f(x)[/latex] and [latex]f(-x)[/latex]
Division Algorithm
given polynomial dividend [latex]f(x)[/latex] and non-zero polynomial divisor [latex]d(x)[/latex] where the degree of [latex]d(x)[/latex] is less than or equal to the degree of [latex]f(x)[/latex], there exist unique polynomials [latex]q(x)[/latex] and [latex]r(x)[/latex] such that [latex]f(x) = d(x)q(x) + r(x)[/latex]
Factor Theorem
[latex]k[/latex] is a zero of polynomial function [latex]f(x)[/latex] if and only if [latex](x - k)[/latex] is a factor of [latex]f(x)[/latex]
Fundamental Theorem of Algebra
a polynomial function with degree greater than 0 has at least one complex zero
imaginary number
a number in the form [latex]bi[/latex] where [latex]i = \sqrt{-1}[/latex]
invertible function
any function that has an inverse function
Linear Factorization Theorem
allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex](x - c)[/latex] where [latex]c[/latex] is a complex number
Rational Zero Theorem
the possible rational zeros of a polynomial function have the form [latex]\frac{p}{q}[/latex] where [latex]p[/latex] is a factor of the constant term and [latex]q[/latex] is a factor of the leading coefficient
Remainder Theorem
if a polynomial [latex]f(x)[/latex] is divided by [latex]x - k[/latex], then the remainder is equal to the value [latex]f(k)[/latex]
synthetic division
a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x - k[/latex]