Question

Which polynomial function could be represented by the graph below?

Answer 1

we know that

Using a graph tool

I'm going to proceed to graph each case to determine the solution

case a) $f(x) = x^3 + x^2 - 6x$

see the attached figure N $1$

case b) $f(x) = x^3 - x^2 - 6x$

see the attached figure N $2$

case c) $f(x) = -2x^3 - 2x^2 +12x$

see the attached figure N $3$

case d) $f(x) = -2x^3 + 2x^2 +12x$

see the attached figure N $4$

therefore

the answer is the option

case c) $f(x) = -2x^3 - 2x^2 +12x$

Answer 2

The polynomial function is $\boxed{f\left( x \right)= - 2{x^3} - 2{x^2} + 12x}$ that is represented by the graph. Option (3) is correct.

Further explanation:

Given:

The options of the equations are as follows.

1.$f\left( x \right) = {x^3} + {x^2} - 6x$

2.$f\left( x \right) = {x^3} - {x^2} - 6x$

3.$f\left( x \right) = - 2{x^3} - 2{x^2} + 12x$

4.$f\left( x \right) = - 2{x^3} + 2{x^2} + 12x$

Explanation:

The graph passes through the points $\left( {-3, 0} \right)$ and $\left( { 2,0} \right).$

Solve the polynomial $f\left( x \right) = {x^3} + {x^2} - 6x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= {x^3} + {x^2} - 6x\\&= x\left({{x^2} + x - 6}\right)\\&= x\left({x - 2} \right)\left( {x + 3} \right)\\\end{aligned}$

The zeros of the polynomial are -3, 0 and 2.

The graph of the polynomial $f\left( x \right) = {x^3} + {x^2} - 6x$ is increasing-decreasing-increasing.

Solve the polynomial $f\left( x \right) = {x^3} - {x^2} - 6x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= {x^3} - {x^2} - 6x\\&= x\left({{x^2} - x - 6} \right)\\&= x\left({x + 2}\right)\left({x - 3}\right)\\\end{aligned}$

The zeros of the polynomial are -2, 0 and 3.

The graph of the polynomial $f\left( x \right) = {x^3} - {x^2} - 6x$ is increasing-decreasing-increasing.

The graph doesn’t passes through the point $\left( { - 3,0} \right).$ Therefore, the polynomial doesn’t satisfy the graph.

Solve the polynomial $f\left( x \right)=- 2{x^3} - 2{x^2}+12x$ to obtain the zeros of x.

$\begin{aligned}f\left( x \right)&= - 2{x^3} + {x^2} + 12x\\&=- 2x\left( {{x^2} + x - 6}\right)\\&=- 2x\left( {x - 2}\right)\left( {x + 3} \right)\\\end{aligned}$

The zeros of the polynomial are -2, 0 and 3.

The graph doesn’t passes through the point $\left({ - 3,0}\right).$ Therefore, the polynomial doesn’t satisfy the graph.

From the graph it has been observed that the graph is decreasing-increasing-decreasing.

The polynomial function is $\boxed{f\left(x\right)= - 2{x^3}- 2{x^2}+12x}$ that is represented by the graph. Option (3) is correct.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

1. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

Answer details:

Grade: High School

Subject: Mathematics

Chapter: polynomials

Keywords: quadratic equation, equation factorization, polynomial, quadratic formula, zeroes, function.