Question

A polynomial ppp has zeros when x=-2x=−2x, equals, minus, 2, x=\dfrac13x= 3 1 ​ x, equals, start fraction, 1, divided by, 3, end fraction, and x=3x=3x, equals, 3. What could be the equation of ppp?

Answer 1

Answer:

Polynomial equation is, $p=(x+2)(3x-1)(x-3)=3x^3-4x^2-17x+6$

Step-by-step explanation:

Given:

The zeros of the polynomial, $p$, are  $x=-2,\frac{1}{3}, 3$.

Therefore,

$x= -2\\(x+2)=0$

So, $(x+2)$ is a factor of the polynomial.

$x=\frac{1}{3}\\3x=1\\3x-1=0$

So, $(3x-1)$ is a factor of the polynomial.

$x=3\\x-3=0$

So, $(x-3)$ is also a factor of the polynomial.

Therefore, a polynomial can be expressed in terms of the product of its factors.

Thus, $p=(x+2)(3x-1)(x-3)$

Expanding $(x+2)\textrm{ and }(3x-1)$, we get

$(x+2)(3x-1)=3x^2-x+6x-2=3x^2+5x-2$

Now, expanding $(3x^2+5x-2)\textrm{ and }(x-3)$, we get

$(3x^2+5x-2)(x-3)=3x^3-9x^2+5x^2-15x-2x+6=3x^3-4x^2-17x+2$

Therefore, the polynomial could be $p=(x+2)(3x-1)(x-3)=3x^3-4x^2-17x+6$