Test Polynomial Functions Wozgen

Front

What is a polynomial function?

Back

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is: $$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial $$7r^3 - 8r^2 + 42r - 48$$, the degree is 3.

Front

How do you factor a polynomial?

Back

To factor a polynomial, you look for common factors, use grouping, or apply special factoring techniques such as difference of squares or perfect square trinomials. For example, $$7r^3 - 8r^2 + 42r - 48$$ can be factored as $$(r^2 + 6)(7r - 8)$$.

Front

What is the range of a polynomial function?

Back

The range of a polynomial function is the set of all possible output values (y-values) that the function can produce. For most polynomial functions, the range is \((-\infty, +\infty)\) unless restricted by specific conditions.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as \(i = \sqrt{-1}\). It is used to express complex numbers, which have a real part and an imaginary part.

Front

How do you multiply complex numbers?

Back

To multiply complex numbers, use the distributive property (FOIL method). For example, to multiply \((3 + 8i)(-2 - i)\), you calculate: \(3(-2) + 3(-i) + 8i(-2) + 8i(-i) = -6 - 3i - 16i - 8 = 2 - 19i\).

Front

What is the sum of two complex numbers?

Back

The sum of two complex numbers is found by adding their real parts and their imaginary parts separately. For example, \((1 + 3i) + (7 - 2i) = (1 + 7) + (3i - 2i) = 8 + i\).

Front

What is a complex conjugate?

Back

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). It is used in various mathematical operations, including division of complex numbers.

Front

What is the difference between a polynomial and a rational function?

Back

A polynomial function is a function that can be expressed as a polynomial, while a rational function is a ratio of two polynomials. For example, \(f(x) = \frac{x^2 + 1}{x - 1}\) is a rational function.

Front

What is the significance of the leading coefficient in a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It affects the end behavior of the polynomial function. For example, in \(7r^3 - 8r^2 + 42r - 48\), the leading coefficient is 7.