Question

question 1
find the degree, leading coefficients, and the maximum number of real zeros of the polynomial.
f(x)=-4x^3 - 4x^2+3x + 5
degree =
leading coefficient =
maximum number of real zeros =
question help: message instructor
submit question
question 2
describe the long run behavior of f(x)=(x - 2)^2(x - 5)^2(x + 2)
as x→-∞, f(x)→∞
as x→∞, f(x)→?

Explanation:

Step1: Identify degree of polynomial

The degree of a polynomial is the highest power of the variable. For $f(x)=-4x^{3}-4x^{2}+3x + 5$, the highest - power of $x$ is 3. So the degree is 3.

Step2: Identify leading coefficient

The leading coefficient is the coefficient of the term with the highest power of the variable. For $f(x)=-4x^{3}-4x^{2}+3x + 5$, the coefficient of $x^{3}$ is - 4.

Step3: Determine maximum number of real zeros

The maximum number of real zeros of a polynomial is equal to its degree. Since the degree of $f(x)=-4x^{3}-4x^{2}+3x + 5$ is 3, the maximum number of real zeros is 3.

for Question 2:

Step1: Expand the polynomial (not necessary for long - run behavior)

The long - run behavior of a polynomial is determined by its leading term. First, expand $f(x)=(x - 2)^{2}(x - 5)^{2}(x + 2)$. $(x - 2)^{2}=x^{2}-4x + 4$, $(x - 5)^{2}=x^{2}-10x + 25$. Then $(x^{2}-4x + 4)(x^{2}-10x + 25)=x^{4}-10x^{3}+25x^{2}-4x^{3}+40x^{2}-100x + 4x^{2}-40x + 100=x^{4}-14x^{3}+69x^{2}-140x + 100$. And $(x^{4}-14x^{3}+69x^{2}-140x + 100)(x + 2)=x^{5}-14x^{4}+69x^{3}-140x^{2}+100x+2x^{4}-28x^{3}+138x^{2}-280x + 200=x^{5}-12x^{4}+41x^{3}-2x^{2}-180x + 200$. The leading term is $x^{5}$ with a positive leading coefficient 1.

Step2: Analyze long - run behavior

As $x\to-\infty$, for a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ odd and $a_n>0$, $y\to-\infty$. But we made a mistake above. We can also analyze without full expansion. The product of the leading terms of $(x - 2)^{2}(x - 5)^{2}(x + 2)$: The leading term of $(x - 2)^{2}$ is $x^{2}$, the leading term of $(x - 5)^{2}$ is $x^{2}$, and the leading term of $(x + 2)$ is $x$. So the leading term of the product is $x^{2}\cdot x^{2}\cdot x=x^{5}$ (positive leading coefficient). As $x\to-\infty$, $f(x)\to-\infty$; as $x\to\infty$, $f(x)\to\infty$.

Answer:

Degree = 3
Leading Coefficient = -4
Maximum number of real zeros = 3