Question

How do you write a second-degree polynomial, with zeroes of -2 and 3, and goes to \[-\infty \] as \[x\to -\infty \]?

Answer 1

Hint: To solve the given question, we will use the given data about the function and then write the function according to it. We know that the degree of the function is the highest power to which the variable is raised, the question asks to write a second-degree polynomial. So, we have to write a quadratic equation with the given data. A quadratic equation has two roots or zeros, say a and b are the roots of a quadratic equation. The simplest quadratic equation we can write is \[\left( x-a \right)\left( x-b \right)\]. Also, if a quadratic equation tends to \[-\infty \] as \[x\to -\infty \] it means that the coefficient of the square term is negative.

Complete step by step solution:
We are asked to write a second-degree polynomial, with zeroes of -2 and 3, and goes to \[-\infty \] as \[x\to -\infty \]. We know that if a and b are the roots of a quadratic equation. The simplest quadratic equation we can write is \[\left( x-a \right)\left( x-b \right)\]. We are already given two zeroes as -2 and 3, using the above method, we can write the required equation as \[\left( x-(-2) \right)\left( x-3 \right)\]. Simplifying the above equation, we get
\[\Rightarrow \left( x+2 \right)\left( x-3 \right)\]
As we are also given that the equation tends to \[-\infty \] as \[x\to -\infty \]. For this condition to be true the coefficient of the square term is negative.
Thus, we get the required equation as
\[\Rightarrow -1\times \left( x+2 \right)\left( x-3 \right)\]
\[\Rightarrow -\left( x+2 \right)\left( x-3 \right)\]

Note: To solve the given equation, we should know the properties of polynomial function such as their degree, roots etc. We can also write the equation by simplifying as \[-{{x}^{2}}+x+6\]. We may need to do calculation in some cases, so mistakes should be avoided.