Certainly! To write a polynomial function in the format [tex]\( y(x) = a\left(x-r_1\right)\left(x-r_2\right)\left(x-r_3\right) \ldots \)[/tex]:
1. Identify the roots (zeroes) of the polynomial: These are the values of [tex]\(x\)[/tex] where the polynomial equals zero.
2. Determine the leading coefficient: This is the constant [tex]\(a\)[/tex] that scales the polynomial.
3. Construct the factored form of the polynomial: This is done by expressing each root [tex]\((r_1, r_2, r_3, \ldots)\)[/tex] as a factor in the form [tex]\((x - r_i)\)[/tex].
Given the roots [tex]\(r_1 = 1\)[/tex], [tex]\(r_2 = -2\)[/tex], and [tex]\(r_3 = 3\)[/tex] with a leading coefficient [tex]\(a = 1\)[/tex], the polynomial [tex]\( f(x) \)[/tex] is expressed as:
[tex]\[
y(x) = 1(x - 1)(x - (-2))(x - 3)
\][/tex]
Simplifying the terms inside the factors, we get:
[tex]\[
y(x) = (x - 1)(x + 2)(x - 3)
\][/tex]
So, the final equation in the factored form is:
[tex]\[
y(x) = (x - 1)(x + 2)(x - 3)
\][/tex]
