Respuesta :
The polynomial function is [tex]\boxed{f\left( 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]f\left( x \right) = {x^3} + {x^2} - 6x[/tex]
2.[tex]f\left( x \right) = {x^3} - {x^2} - 6x[/tex]
3.[tex]f\left( x \right) = - 2{x^3} - 2{x^2} + 12x[/tex]
4.[tex]f\left( 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]f\left( x \right) = {x^3} + {x^2} - 6x[/tex] to obtain the zeros of x.
[tex]\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}[/tex]
The zeros of the polynomial are -3, 0 and 2.
The graph of the polynomial [tex]f\left( x \right) = {x^3} + {x^2} - 6x[/tex] is increasing-decreasing-increasing.
Solve the polynomial [tex]f\left( x \right) = {x^3} - {x^2} - 6x[/tex] to obtain the zeros of x.
[tex]\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}[/tex]
The zeros of the polynomial are -2, 0 and 3.
The graph of the polynomial [tex]f\left( 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]f\left( x \right)=- 2{x^3} - 2{x^2}+12x[/tex] to obtain the zeros of x.
[tex]\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}[/tex]
The zeros of the polynomial are -2, 0 and 3.
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{f\left(x\right)= - 2{x^3}- 2{x^2}+12x}[/tex] that is represented by the graph. Option (3) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: polynomials
Keywords: quadratic equation, equation factorization, polynomial, quadratic formula, zeroes, function.
The polynomial function could be represented by the graph below is [tex]\rm f(x) = x^3 + x^2 - 6x[/tex].
Thus, the correct option is A.
We have to determine
Which polynomial function could be represented by the graph below?
What is polynomial?
The expression function is an expression of more than two algebraic terms with constant exponents.
In the given graph the graph intersects at three points the zeros of the polynomial are -3, 0, and 2.
Therefore,
The polynomial function could be represented by the graph below is;
[tex]\rm (x+3)(x-0)(x-2)=0\\\\(x^2+3x)(x-2)=0\\\\x^3-2x^2+3x^2-6x=0\\\\ x^3 + x^2- 6x=0[/tex]
Hence, the polynomial function could be represented by the graph below is [tex]\rm f(x) = x^3 + x^2 - 6x[/tex].
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