Understanding Polynomial Functions: Video e Problemi di Pratica

For the following polynomial function, use the Leading Coefficient Test to determine the end behavior of the graph and sketch the graph of the function.

f(x) = -7x³ + 2x² - 4x

For the following polynomial function, use the Leading Coefficient Test to determine the end behavior of the graph and sketch the graph of the function.

f(x) = 2x⁶ + 3x⁴ - 5x² + 1

Graph the given function: f(x) = x³ + 2x² - 3x - 4

(i) Determine the graph's end behavior using the Leading Coefficient Test

(ii) Determine if it has y-axis symmetry, origin symmetry, or neither

(iii) Graph

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: f(x) = 7x³ + 2x² - 3x + 11

Graph the given function.

(i) Determine the graph's end behavior using the Leading Coefficient Test

(ii) Determine if it has y-axis symmetry, origin symmetry, or neither

(iii) Graph

f(x) = - 2x(2x² - 3)

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: f(x) = 3x⁴ - 3x² - x + 5

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: f(x) = -9x⁴ + 9x² + 2x + 11

Graph the given polynomial expression. f(x) = x²(x + 1)³(x - 2)

Identify the zeros of the given polynomial function and state their multiplicities. Describe also how the graph behaves with the x-axis.

f(x) = 5(x - 3)(x + 6)²

Identify the zeros of the given polynomial function and state their multiplicities. Describe also how the graph behaves with the x-axis.

f(x) = 5(x + 6)(x + 8)²

Graph the given polynomial function. f(x) = -(1/2)x³(x + 3)²(2x - 2)

Identify the zeros of the given polynomial function and state their multiplicities. Describe also how the graph behaves with the x-axis. 

f(x) = -7(x + 3/5)(x - 9)³

Identify the zeros of the given polynomial function and state their multiplicities. Describe also how the graph behaves with the x-axis.

f(x) = x³ + 8x² - 9x - 72

Justify that the polynomial f(x) = x³ + 4x - 3 has a real zero in between 0 and 1 by using the Intermediate Value Theorem.

Justify that the polynomial f(x) = 2x³ - 8x² + 4 has a real zero in between -1 and 0 by using the Intermediate Value Theorem.

Justify that the polynomial f(x) = 2x⁴ + 5x³ - 9x² has a real zero in between 1 and 2 by using the Intermediate Value Theorem.

Justify that the polynomial f(x) = 2x³ + 4x² - 6x + 3 has a real zero in between -4 and -3 by using the Intermediate Value Theorem.

Justify that the polynomial f(x) = 5x³ - 3x² + 2x + 1 has a real zero in between -1 and 0 by using the Intermediate Value Theorem.

Factor completely:

x³ + 4x² -25x - 100

Consider the following function: f(x) = 4x⁴ - 7x² + 1

Determine if it is even, odd, or neither; and describe the symmetry, if any.

How many of the following are polynomial functions?

$f\left(x\right)=3x^{\frac{5}{2}}-x+1$

$g\left(x\right)=-\frac{2}{3}{x}^4+7x^2-9$

$h\left(x\right)=x^3+4x^{-2}-1$

$p\left(x\right)=5x^7-2x^3+x$

Let $f\left(x\right)=(3x^2-5x+1)+(4x^3-2x^2+7)-x^3$. After writing $f$ in standard form, find its leading coefficient.

For $f\left(x\right)=-7x^8+3x^5-2$, compute $\underset{x\to \infty }{\lim }f\left(x\right)$.

Let $f\left(x\right)=5x^7-2x+1$. Compute $\underset{x\to -\infty }{\lim }f\left(x\right)$.

For $f\left(x\right)=(x-2{)}^4(x+5{)}^3$, find the multiplicity of the zero at $x=-5$.

A graph is claimed to represent a polynomial with $7$ turning points. What is the least possible degree of such a polynomial?

Let $f\left(x\right)=-3x^5+2x^2-1$. Evaluate $\underset{x\to -\infty }{\lim }f\left(x\right)$.

For $f\left(x\right)=(x+2{)}^2(x-1{)}^3$, determine the multiplicity of the zero at $x=1$.

A polynomial has degree $8$. What is the maximum possible number of turning points?