Chapter 5

What is true of the appearance of graphs that reflect a direct variation between two variables?

Explain why we cannot find inverse functions for all polynomial functions.

What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?

Describe a use for the Remainder Theorem.

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

What is the difference between an \(x\) - intercept and a zero of a polynomial function \(f ?\)

Explain the difference between the coefficient of a power function and its degree.

Explain the advantage of writing a quadratic function in standard form.

If two variables vary inversely, what will an equation representing their relationship look like?

What is the fundamental difference in the graphs of polynomial functions and rational functions?

Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.

If a polynomial of degree \(n\) is divided by a binomial of degree \(1,\) what is the degree of the quotient?

If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

Is there a limit to the number of variables that can vary jointly? Explain.

When finding the inverse of a radical function, what restriction will we need to make?

If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?

What is the difference between rational and real zeros?

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(x^{2}+5 x-1\right) \div(x-1) $$

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

Explain why the condition of \(a \neq 0\) is imposed in the definition of the quadratic function.

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as \(x\) and when \(x=6, y=12\).

Can a graph of a rational function have no vertical asymptote? If so, how?

If Descartes' Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{2}-9 x-5\right) \div(x-5) $$

Explain how the factored form of the polynomial helps us in graphing it.

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

What is another name for the standard form of a quadratic function?

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square of \(x\) and when \(x=4, \quad y=80\).

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x-4)^{2},[4, \infty) $$

Can a graph of a rational function have no \(x\) -intercepts? If so, how?

If synthetic division reveals a zero, why should we try that value again as a possible solution?

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(3 x^{2}+23 x+14\right) \div(x+7) $$

If the graph of a polynomial just touches the \(x\) -axis and then changes direction, what can we conclude about the factored form of the polynomial?

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As \(x \rightarrow-\infty, \quad f(x) \rightarrow-\infty\) and as \(x \rightarrow \infty, \quad f(x) \rightarrow-\infty\).

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the square root of \(x\) and when \(x=36, \quad y=24\).

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x+2)^{2},[-2, \infty) $$

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x-1}{x+2} $$

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(x^{4}-9 x^{2}+14\right) \div(x-2) $$

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(4 x^{2}-10 x+6\right) \div(4 x+2) $$

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ C(t)=2(t-4)(t+1)(t-6) $$

For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ f(x)=x^{5} $$

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ f(x)=x^{2}-12 x+32 $$

For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies directly as the cube of \(x\) and when \(x=36, \quad y=24\).

For the following exercises, find the inverse of the function on the given domain. $$ f(x)=(x+1)^{2}-3,[-1, \infty) $$

For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x+1}{x^{2}-1} $$

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(3 x^{3}-2 x^{2}+x-4\right) \div(x+3) $$

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(6 x^{2}-25 x-25\right) \div(6 x+5) $$