Glossary
Complex numbers
Numbers that combine a real part and an imaginary part, written in the form *a + bi*, where *a* and *b* are real numbers.
Example:
A physicist might use complex numbers to describe wave functions in quantum mechanics.
Conjugate pairs
A property of complex zeros of polynomials with real coefficients, stating that if *a + bi* is a zero, then its conjugate *a - bi* must also be a zero.
Example:
If 2 + 3i is a zero of a polynomial, then 2 - 3i must also be part of its conjugate pairs.
Degree (of a polynomial)
The highest power of the variable in a polynomial function, which also indicates the total number of complex zeros (counting multiplicities).
Example:
A polynomial modeling the path of a thrown object is typically of degree 2, representing a parabolic trajectory.
Even functions
Functions that are symmetric about the y-axis, satisfying the property f(-x) = f(x).
Example:
The path of a perfectly symmetrical arch bridge can be modeled by an even function.
Even multiplicity
When a real zero has an even multiplicity, the graph of the polynomial touches the x-axis at that point but does not cross it, indicating no sign change.
Example:
If a polynomial has a zero at x=4 with even multiplicity, its graph will bounce off the x-axis at x=4, like a parabola's vertex.
Imaginary numbers
Numbers involving the imaginary unit *i*, where *i* = √-1, typically written in the form *bi* where *b* is a real number.
Example:
In electrical engineering, imaginary numbers are used to represent the phase of an alternating current.
Imaginary unit i
The fundamental unit of imaginary numbers, defined as the square root of -1 (i = √-1).
Example:
Solving the equation x² = -4 requires understanding the imaginary unit i, as x would be ±2i.
Linear factor
An expression of the form (x - a), where *a* is a zero of a polynomial function. If *a* is a real zero, (x - a) is a linear factor of *p(x)*.
Example:
For the polynomial (x-5)(x+2), (x-5) is a linear factor corresponding to the zero x=5.
Multiplicity
The number of times a linear factor (x - a) appears in the factored form of a polynomial, indicating how many times *a* is a zero.
Example:
In the polynomial (x-1)²(x+3), the zero x=1 has a multiplicity of 2, meaning its factor appears twice.
Odd functions
Functions that are rotationally symmetric about the origin, satisfying the property f(-x) = -f(x).
Example:
The graph of a roller coaster's initial drop and subsequent rise might exhibit characteristics of an odd function if it's symmetric about the origin.
Real numbers
All numbers that can be found on a number line, including integers, fractions, decimals, and irrational numbers.
Example:
When calculating the trajectory of a rocket, you'd use real numbers to represent its height and distance over time.
Successive differences
A method used to determine the degree of a polynomial by repeatedly calculating the differences between consecutive output values until a constant difference is found.
Example:
To confirm if a data set represents a quadratic relationship, you would check if the second layer of successive differences is constant.
X-intercepts
The points where the graph of a polynomial function crosses or touches the x-axis, corresponding to the real zeros of the function.
Example:
On a graph showing the height of a ball over time, the x-intercepts would represent the moments the ball hits the ground.
Zero (of a polynomial function)
A value *a* for which a polynomial function *p(x)* equals zero, meaning *p(a) = 0*. Also known as a root.
Example:
If a polynomial models the profit of a company, the zeros would indicate the break-even points where profit is zero.