Graph each of the following polynomials on your own piece of paper or using graphing software like Desmos. What is similar about them? What is different? For each of these questions, select all that apply.
a) $f(x)=4x^4+5x-4$
b) $g(x)=4x^4+2x$
c) $h(x)=4x^4+6$

What is similar?
A. For each, as $x$ goes to $+\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$.
B. They are all degree 4 polynomials.
C. For each, as $x$ goes to $-\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$.
D. They all have a leading coefficient of 4 .
E. For each, as $x$ goes to $-\mathrm{\infty }$, the function goes to $-\mathrm{\infty }$.
F. For each, as $x$ goes to $+\mathrm{\infty }$, the function goes to $-\mathrm{\infty }$.
G. They all have 2 terms
H. They all have only one $x$-intercept.

What is different?
A. They have a different number of terms.
B. They have different $x$-intercepts.
C. They have different $y$ intercepts.
D. They have different leading coefficients.
E. They are different degree polynomials.
F. They have different long-run behaviors.

Step 1 ：Given the polynomials $f(x)=4x^4+5x-4$, $g(x)=4x^4+2x$, and $h(x)=4x^4+6$.

Step 2 ：Identify the degree of the polynomials. The degree of a polynomial is the highest power of x in its expression. All three polynomials are of degree 4.

Step 3 ：Identify the leading coefficient of the polynomials. The leading coefficient is the coefficient of the term with the highest power. All three polynomials have a leading coefficient of 4.

Step 4 ：Identify the long-run behavior of the polynomials. The long-run behavior of a polynomial refers to the behavior of the polynomial as x approaches positive or negative infinity. For all three polynomials, as $x$ goes to $+\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$ and as $x$ goes to $-\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$.

Step 5 ：Identify the number of terms in each polynomial. The polynomials $f(x)$, $g(x)$, and $h(x)$ have 3, 2, and 2 terms respectively.

Step 6 ：Identify the x-intercepts of the polynomials. The x-intercepts are the values of x for which the polynomial equals zero. The x-intercepts for the polynomials are complex numbers, which means they do not intersect the x-axis in the real number plane. However, the number of x-intercepts (including complex ones) is different for each polynomial.

Step 7 ：Identify the y-intercepts of the polynomials. The y-intercept is the value of the polynomial when x=0. The y-intercepts for the polynomials $f(x)$, $g(x)$, and $h(x)$ are -4, 0, and 6 respectively.

Step 8 ：From the above analysis, we can conclude that the similarities between the polynomials are: For each, as $x$ goes to $+\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$. They are all degree 4 polynomials. For each, as $x$ goes to $-\mathrm{\infty }$, the function goes to $+\mathrm{\infty }$. They all have a leading coefficient of 4.

Step 9 ：The differences between the polynomials are: They have a different number of terms. They have different $x$-intercepts. They have different $y$ intercepts.

Step 10 ：$\boxed{{\displaystyle \text{Similarities:}}}$ $\boxed{{\displaystyle \text{A. For each, as }x\text{ goes to }+\mathrm{\infty },\text{ the function goes to }+\mathrm{\infty }.}}$ $\boxed{{\displaystyle \text{B. They are all degree 4 polynomials.}}}$ $\boxed{{\displaystyle \text{C. For each, as }x\text{ goes to }-\mathrm{\infty },\text{ the function goes to }+\mathrm{\infty }.}}$ $\boxed{{\displaystyle \text{D. They all have a leading coefficient of 4.}}}$

Step 11 ：$\boxed{{\displaystyle \text{Differences:}}}$ $\boxed{{\displaystyle \text{A. They have a different number of terms.}}}$ $\boxed{{\displaystyle \text{B. They have different }x\text{-intercepts.}}}$ $\boxed{{\displaystyle \text{C. They have different }y\text{ intercepts.}}}$