Polynomial Functions and Their Properties
Polynomial functions are construct that are easy to comprehend but hard to understand fully. It involves unknown variables that are combined only with addition, multiplication, and exponentiation to positive. For example, a general form of a polynomial function is
$f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$
where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
The degree of a polynomial is the highest power of the variable in the polynomial.
The graph of a polynomial function can provide valuable insights into its behavior:
- The y-intercept is the point where the graph crosses the y-axis, given by $f(0)$.
- The x-intercepts, also known as zeros or roots, are the points where the graph crosses the x-axis.
Consider the polynomial $f(x) = x^3 - 4x^2 + 3x + 2$
- It's a cubic polynomial (degree 3)
- The y-intercept is $f(0) = 2$
- To find the x-intercepts, we need to solve $f(x) = 0$
Factor Theorem
The Factor Theorem states that a polynomial $f(x)$ has a factor $(x - a)$ if and only if $f(a) = 0$.
In other words, $a$ is a root of the polynomial if and only if $(x - a)$ is a factor of the polynomial.
This theorem provides a way to find factors of a polynomial by testing potential roots.
For $f(x) = x^3 - 4x^2 + 3x + 2$, let's check if $x = 1$ is a root: $f(1) = 1^3 - 4(1)^2 + 3(1) + 2 = 1 - 4 + 3 + 2 = 2$ Since $f(1) \neq 0$, $(x - 1)$ is not a factor of $f(x)$.
Remainder Theorem
The Remainder Theorem is closely related to the Factor Theorem. It states that the remainder of a polynomial $f(x)$ divided by $(x - a)$ is equal to $f(a)$.
A function $f(x) can be written as
$$f(x) = g(x)(x-a) + R$$
where $g(x)$ is another polynomial and $R$ is some real constant.
Dividing both sides by $x-a$ shows that
$$\frac{f(x)}{x-a} = g(x)+ \frac{R}{x-a}$$
so $R$ is the remainder. When $a$ is subbed into the first expression
$$f(a) = g(a)(a-a) + R$$
$$f(a) = R$$
so $f(a)$ is the remainder.
This theorem is particularly useful for finding remainders without performing long division.
For $f(x) = x^3 - 4x^2 + 3x + 2$, find the remainder when divided by $(x - 2)$: $f(2) = 2^3 - 4(2)^2 + 3(2) + 2 = 8 - 16 + 6 + 2 = 0$ The remainder is 0, which means $(x - 2)$ is a factor of $f(x)$.
Polynomial Division
When doing division of polynomials by an integer root $(x-a)$, you do it basically the same way as normal division. Easier to show with a graph.
Sum and Product of Roots
For a polynomial equation $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 = 0$, you can relate the coefficients and the roots by:
- Sum of the roots: $S = -(a_{n-1}/a_n)$
- Product of the roots: $P = (-1)^n(a_0/a_n)$
These formulas are particularly useful when working with polynomials of higher degrees.
For the quadratic equation $x^2 + 5x + 6 = 0$:
- Sum of roots: $S = -5/1 = -5$
- Product of roots: $P = (-1)^2(6/1) = 6$
Indeed, the roots are -2 and -3, and $(-2) + (-3) = -5$ and $(-2)(-3) = 6$.
Students often forget to consider the sign when calculating the sum and product of roots. Remember that the sum formula has a negative sign, while the product formula includes $(-1)^n$.
