write an expression in factored form for the polynomial of least possible degree graphed below. y(x) =
write an expression in factored form for the polynomial of least possible degree graphed below. y(x) =
Answer
Explanation:
Step1: Identify x - intercepts
The x - intercepts of the polynomial graph are (x=-2), (x = 1), and (x=3). A polynomial (y(x)) with x - intercepts (x = a_1,a_2,\cdots,a_n) can be written in factored form as (y(x)=k(x - a_1)(x - a_2)\cdots(x - a_n)), where (k) is a non - zero constant. So, a first - guess at the factored form is (y(x)=k(x + 2)(x - 1)(x - 3)).
Step2: Determine the leading coefficient (k)
We can use a point on the graph, say the y - intercept. When (x = 0), from the graph (y(0)=- 4). Substitute (x = 0) into (y(x)=k(x + 2)(x - 1)(x - 3)): [y(0)=k(0 + 2)(0 - 1)(0 - 3)] [y(0)=k(2)(-1)(-3)=6k] Since (y(0)=-4), then (6k=-4), and (k=-\frac{2}{3}).
Answer:
(y(x)=-\frac{2}{3}(x + 2)(x - 1)(x - 3))