a)Degree is 4, leading coefficient is 1 and end behavior is As x→ -∞ ,y → ∞ As x→ ∞, y → ∞
b)Zero at x = 1 with multiplicity 1(crosses)
Zero at x = 3 with multiplicity 1(crosses)
Zero at x = -2 with multiplicity 2(touches)
c) y-intercept is 12
How to plot graph of polynomial function.
Given polynomial function f(x) = (x-1)(x-3)(x+2)²
a) The degree of the polynomial is determined by the highest power of x which is the exponent 2 in (x+2)²
The degree = 2*2 = 4
The leading coefficient = 1 (coefficient of term with highest power)
b) For even-degree polynomials with a positive leading coefficient, the end behavior as x approaches positive or negative infinity is upward (both ends rise).
As x→ -∞ ,y → ∞ As x→ ∞, y → ∞
Zeros are the values of x that make f(x) = 0. The points at which the curve crosses or touches the x-axis. Multiplicity means the number of times it occurs
Zero at x = 1 with multiplicity 1
Zero at x = 3 with multiplicity 1
Zero at x = -2 with multiplicity 2
From the factored form (x+2) is raised to 2
Zeros with odd multiplicities correspond to roots that cross the x-axis.
At x = 1 and 3, the curve crossed the x-axis.
Zeros with even multiplicities correspond to roots that touch the x-axis.
At x = -2, the curve touches the x-axis.
c) The y-intercept is found by setting x = 0.
f(0) = (0-1)(0-3)(0+2)²
= -1*-3*2²
= 12