Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s). the degree of the function $f(x) = -(x + 1)^2(2x - 3)(x + 2)^2$ is \boxed{ }, and its $y$-intercept is ( \boxed{ } , \boxed{ } ).

Explanation:

Step1: Find the degree of the polynomial

To find the degree of a polynomial in factored form, we add the exponents of each factor. The factors are \((x + 1)^2\) (exponent 2), \((2x - 3)\) (exponent 1, since it's to the first power), and \((x + 2)^2\) (exponent 2). So we calculate \(2+1 + 2\).
\(2+1+2 = 5\).

Step2: Find the y - intercept

The y - intercept of a function \(y = f(x)\) is found by setting \(x = 0\) and solving for \(y\) (or \(f(0)\)).
Substitute \(x = 0\) into the function \(f(x)=-(x + 1)^2(2x - 3)(x + 2)^2\):
First, calculate each part:
\((0 + 1)^2=1^2 = 1\)
\(2(0)-3=-3\)
\((0 + 2)^2=2^2 = 4\)
Then, \(f(0)=-(1)\times(-3)\times(4)\)
\(f(0)=-1\times(-3)\times4 = 12\)
So the y - intercept is at the point \((0,12)\).

Answer:

The degree of the function is \(5\), and its y - intercept is \((0,12)\). So the answers are \(5\), \(0\), and \(12\) (in the respective boxes).