Which polynomial function could be represented by the graph below? f(x) = x3 + x2 – 6x f(x) = x3 – x2 – 6x f(

Question

Which polynomial function could be represented by the graph below?

f(x) = x3 + x2 – 6x
f(x) = x3 – x2 – 6x
f(x) = -2×3 – 2×2 + 12x
f(x) = -2×3 + 2×2 + 12x
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1. The polynomial function is [tex]boxed{fleft( x right) = – 2{x^3} – 2{x^2} + 12x}[/tex] that is represented by the graph. Option (3) is correct.

Further explanation:

Given:

The options of the equations are as follows.

1.[tex]fleft( x right) = {x^3} + {x^2} – 6x[/tex]

2. [tex]fleft( x right) = {x^3} – {x^2} – 6x[/tex]

3. [tex]fleft( x right) = – 2{x^3} – 2{x^2} + 12x[/tex]

4. [tex]fleft( x right) = – 2{x^3} + 2{x^2} + 12x[/tex]

Explanation:

The graph passes through the points [tex]left( {-3, 0}right)[/tex] and [tex]left( { 2,0} right).[/tex]

Solve the polynomial [tex]fleft( x right) = {x^3} + {x^2} – 6x[/tex] to obtain the zeros of x.

[tex]begin{aligned}fleft( x right)&= {x^3} + {x^2} – 6x&= xleft({{x^2} + x – 6}right)&= xleft({x – 2}right)left({x + 3}right)end{aligned}[/tex]

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

The graph of the polynomial [tex]fleft( x right) = {x^3} + {x^2} – 6x[/tex] is increasing-decreasing-increasing.

Solve the polynomial [tex]fleft( x right) = {x^3} – {x^2} – 6x[/tex] to obtain the zeros of x.

[tex]begin{aligned}fleft( x right)&={x^3} – {x^2} – 6x&= xleft({{x^2} – x – 6}right)&= xleft({x + 2} right)left({x – 3} right)end{aligned}[/tex]

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

The graph of the polynomial [tex]fleft( x right) = {x^3} – {x^2} – 6x[/tex] is increasing-decreasing-increasing.

The graph doesn’t passes through the point [tex]left({ – 3,0} right).[/tex]Therefore, the polynomial doesn’t satisfy the graph.

Solve the polynomial [tex]fleft( x right)= – 2{x^3} – 2{x^2} + 12x[/tex] to obtain the zeros of x.

[tex]begin{aligned}fleft( x right)&= – 2{x^3} + {x^2} + 12x&= – 2xleft({{x^2} + x – 6} right)&= – 2xleft( {x – 2}right)left({x + 3}right)end{aligned}[/tex]

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

The graph of the polynomial [tex]fleft( x right)=- 2{x^3} – 2{x^2} + 12x[/tex] is decreasing-increasing-decreasing.

Solve the polynomial [tex]fleft( x right)= – 2{x^3} + 2{x^2} + 12x[/tex] to obtain the zeros of x.

[tex]begin{aligned}fleft( x right)&= – 2{x^3} + 2{x^2} + 12x&=- 2xleft( {{x^2} – x – 6} right)&=- 2xleft({x + 2} right)left({x – 3} right)end{aligned}[/tex]

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

The graph of the polynomial [tex]fleft( x right) = – 2{x^3} + 2{x^2} + 12x[/tex] is decreasing-increasing-decreasing.

The graph doesn’t passes through the point [tex]left({ – 3,0}right).[/tex] 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 [tex]boxed{fleft( x right)= – 2{x^3} – 2{x^2} +12x}[/tex] that is represented by the graph. Option (3) is correct.

Subject: Mathematics

Chapter: polynomials

2. Note that

1.

[tex]f(x)=x^3+x^2-6x=x(x^2+x-6)=x(x-2)(x+3)[/tex]

The x-intercepts are at points x=-3, x=0, x=2. The graph should be increasing – decreasing – increasing.

2.

[tex]f(x)=x^3-x^2-6x=x(x^2-x-6)=x(x+2)(x-3)[/tex]

The x-intercepts are at points x=-2, x=0, x=3. The graph should be increasing – decreasing – increasing.

3.

[tex]f(x)=-2x^3-2x^2+12x=-2x(x^2+x-6)=-2x(x-2)(x+3)[/tex]

The x-intercepts are at points x=-3, x=0, x=2. The graph should be decreasing – increasing – decreasing.

4.

[tex]f(x)=-2x^3+x^2+12x=-2x(x^2-x-6)=-2x(x+2)(x-3)[/tex]

The x-intercepts are at points x=-2, x=0, x=3. The graph should be decreasing – increasing – decreasing.

From the diagram you can see that x-intercepts are at points x=-3, x=0, x=2 and the graph is decreasing-increasing-decreasing.