Give the constant term, $a_0$.

$$15-11 t^9-8 t^4$$

Which of the expression are polynomials in $x$? If an expression is not a polynomial in $x$, what rules it out?

$$\frac{2 x^3}{5}-2 x^7$$

Pascal's triangle ${ }^2$ is an arrangement of integers that begins

$$\begin{aligned} &\\&\\ &\begin{array}{cccccccccc} & & & & 1& & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array} \end{aligned}$$

with $n$ entries in each row formed by adding the two entries above it. For instance, the boldface 3 above is formed by adding the preceding entries 1 and 2 , and the boldface 4 is formed by adding the preceding entries 3 and 1. Given this information, Without multiplying it out, write $(x+1)^7$ in standard form. [Hint: Make an educated guess based on your answers to Problems 67 and 68.]

The coefficient of $t$ in $t\left(a+(t+1)^{10}\right)$ is zero, give the value of $a$ that makes the statement true.

Pascal's triangle ${ }^2$ is an arrangement of integers that begins

$$\begin{aligned} &\\&\\ &\begin{array}{cccccccccc} & & & & 1& & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array} \end{aligned}$$

with $n$ entries in each row formed by adding the two entries above it. For instance, the boldface 3 above is formed by adding the preceding entries 1 and 2 , and the boldface 4 is formed by adding the preceding entries 3 and 1. Given this information, Write the following polynomials in standard form:

$$(x+1)^0, \quad(x+1)^1, \quad(x+1)^2, \quad(x+1)^3 .$$

What do you notice?

Pascal's triangle ${ }^2$ is an arrangement of integers that begins

$$\begin{aligned} &\\&\\ &\begin{array}{cccccccccc} & & & & 1& & & & \\ & & & 1 & & 1 & & & \\ & & 1 & & 2 & & 1 & & \\ & 1 & & 3 & & 3 & & 1 & \\ 1 & & 4 & & 6 & & 4 & & 1 \end{array} \end{aligned}$$

with $n$ entries in each row formed by adding the two entries above it. For instance, the boldface 3 above is formed by adding the preceding entries 1 and 2 , and the boldface 4 is formed by adding the preceding entries 3 and 1. Given this information, Find the next 3 rows of Pascal's triangle.

Using a calculator or computer, sketch the graph of $y=(x-a)^3+3 a(x-a)^2+3 a^2(x-a)+a^3$ for $a=0, a=1$, and $a=2$. What do you observe? Use algebra to confirm your observation.

Does the polynomial function $y=a x^4+2 x^3-9 x^2+$ $3 x-18$, with $a>0$, have any zeros? If so, at least how many, and how do you know? If not, why not?

The constant term of $(t-1)^5-(t-a)$ is zero, give the value of $a$ that makes the statement true.

The degree of $(t-1)^3+a(t+1)^3$ is less than 3, give the value of $a$ that makes the statement true.

If $a$ is a nonzero constant, for what values of $x$ is $x^3-a x^2+a^2 x-a^3$ positive?

Consider the polynomial $g(t)=\left(t^2+5\right)(t-a)$, where $a$ is a constant. For which value(s) of $t$ is $g(t)$ positive?

A polynomial whose product with itself has degree 9, give polynomials satisfying the given conditions if possible, or say why it is impossible to do so.

Two polynomials whose product has degree 9, give polynomials satisfying the given conditions if possible, or say why it is impossible to do so.