Determine whether the following are polynomials:
A \left( x \right) = 4 x^{\frac{1}{4}} + 2 x^{5} + 2
3 x^{3} + \dfrac{2}{x^{7}} - 1
Consider the expression x + 7 x^{6} - x^{9}.
How many terms are in the expression?
What are the coefficients of the terms?
True or false: The degree of a polynomial in x is the largest coefficient of any of the terms of the polynomial.
For each of the following polynomials:
P \left( x \right) = 2 x^{7} + 2 x^{5} + 2 x + 2
P \left(x\right) = 7 \sqrt{6} - \sqrt{5} x^{5} + 5
P \left( x \right) = \dfrac{x^{7}}{5} + \dfrac{x^{6}}{6} + 5
Is it possible for the graph of a polynomial function to have:
No x-intercepts?
No y-intercepts?
State whether the following functions have exactly two x-intercepts.
y = \left(x + 3\right)^{3}
y = \left(x + 6\right)^{2} \left(x + 5\right)
y = \left(x + 7\right) \left(x - 1\right) \left(x - 4\right)
y = x \left(x - 2\right) \left(x - 8\right)
Patricia claims she can graph a third-degree polynomial function with 3 turning points.
Is this possible or impossible?
Consider the function f \left( x \right) = x^{7} - 9 x^{3} - 2.
What is the maximum number of real zeros that the function can have?
What is the maximum number of x-intercepts that the graph of the function can have?
What is the maximum number of turning points that the graph of the function can have?
For the polynomial P(x)= 4 - \dfrac{7 x^{6}}{6}, find:
P(-1)
P(\sqrt{2})
P\left(\dfrac{1}{2}\right)
Consider the function f \left( x \right) = x^{4} - 7 x^{3} + 12 x^{2} + 4 x - 16. Determine whether the following is a zero of the function:
x=5
x=4
x=- 1
The polynomial P \left( x \right) = a x^{4} - 6 x^{2} + 2 x + b is a monic polynomial of degree 4, with a constant term of - 2.
Find the value of a.
Find the value of b.
The following graphs are of polynomials of the form y=x^n. For each graph, determine whether n is even or odd:
Consider the functions y = x^{2}, y = x^{4} and y=x^6.
Describe the general shape of the graph of each function.
Sketch the graph of y = x^{2}, y = x^{4} and y=x^6 on the same number plane.
Describe what happens to the graph of a function of the form y=x^{2n} as n increases.
Consider the functions y = x^{3}, y = x^{5}, and y=x^7.
Graph the three functions on the same number plane.
Describe what happens to the graph of a function of the form y=x^{2n+1} as n increases.
Consider the function y = - x^{7}.
As x approaches infinity, what happens to the corresponding y-values?
As x approaches negative infinity, what happens to the corresponding y-values?
Sketch the general shape of y = - x^{7}.
Consider y = x^{3} - 8.
Find the coordinates of the point of inflection on the curve.
Find the x-intercept of the curve.
Hence plot the curve y = x^{3} - 8.
For each of the following functions:
Find the x-intercept(s).
Find the y-intercept(s).
Plot the graph of the curve.
y = \left(x + 3\right) \left(x + 2\right) \left(x - 2\right)
y = \left(x - 2\right)^{2} \left(x + 5\right)
y = \left(x - 1\right) \left(x - 2\right) \left(x + 4\right) \left(x + 5\right)
y = - \left(x + 1\right) \left(x + 3\right) \left(x + 4\right) \left(x - 4\right)
y = \left(x - 2\right)^{2} \left(x + 3\right) \left(x - 1\right)
y = - \left(x - 3\right)^{2} \left(x + 1\right) \left(x - 2\right)
y = - \left(x - 3\right)^{2} \left(x + 2\right)^{2}
y = \left(x + 2\right)^{3} \left(x - 2\right)
For each of the following functions:
As x \to -\infty what does y approach?
As x \to \infty what does y approach?
Find the x-intercepts for this function
Find the y-intercept of the function
Sketch the graph of the function.
Graph the function f \left( x \right) = x \left(x + 3\right) \left(x - 3\right).
Consider the polynomial function y = x^{4} - 4 x^{2}
Determine the leading coefficient.
Does the function rise or fall to the left?
Does the function rise or fall to the right?
Express the equation in factorised form.
Find the x-intercepts.
Find the y-intercept.
Sketch the graph of y = x^{4} - 4 x^{2}.
Consider the polynomial function y = x^{4} - x^{2}
Determine the leading coefficient.
Is the degree odd or even?
Does the function rise or fall to the left?
Does the function rise or fall to the right?
Hence, sketch the graph of y = x^{4} - x^{2}.
Sketch the graph of the function f \left( x \right) = 2 x^{4} - x^{2} + 2.
Consider the curve y = - \left(x - 1\right)^{2} \left(x^{2} - 9\right).
Find the x-intercept(s).
Find the y-intercept(s).
Determine whether the graph has y-axis symmetry, origin symmetry, or neither.
Plot the graph of the curve.
Find the equation for each of the following curves in factored form. Assume each curve is a monic cubic polynomial.
Consider the graph of y = f \left( x \right):
What are the roots of y = f \left( x \right)?
Find the equation of f \left( x \right).
Solve the inequality f \left( x \right) \geq 0.