Question

How to use zero factor property in reverse?

Answer 1

Hint: Zero factor property is defined as:
Consider the equation $a \times b = 0$ that is the product of two unknown numbers is $0$. Then we can find the unknown values by $a = 0$ or $b = 0$.
We can use this reverse of the property to determine the polynomial function. From the zeros of the polynomial we can find the factors. Multiply the factors and equate to zero that gives the polynomial.

Complete step-by-step answer:
Zero factor property is defined as: Consider the equation $a \times b = 0$ that is the product of two unknown numbers is $0$, where $a$ and $b$ are the factor of the equation. Then we can find the unknown values by $a = 0$ or $b = 0$.
Reverse of the zero property is that, if $a$ and $b$ are the zeros then $x - a$ and $x - b$ are the factors of the equation. Multiplying the factors and equating them to zero will give the polynomial.
$ \Rightarrow$$(x - a)(x - b) = 0$
$ \Rightarrow {x^2} - bx - ax + ab = 0$
$ \Rightarrow {x^2} - (a + b)x + ab = 0$
Here, we took two zeros therefore the polynomial is the quadratic equation.
This will give us the polynomial.
For example, suppose we have the zeros: -2, 3, and 1.
That means,
$ \Rightarrow$$x = - 2$
$ \Rightarrow x + 2 = 0 \ldots (1)$
$ \Rightarrow$$x = 3$
$ \Rightarrow x - 3 = 0 \ldots (2)$
$ \Rightarrow$$x = 1$
$ \Rightarrow x - 1 = 0 \ldots (3)$
Multiply the factors given in the equation $(1),(2)$ and $(3)$.
$ \Rightarrow$$(x + 2)(x - 3)(x - 1) = 0$
$ \Rightarrow ({x^2} - x - 6)(x - 1) = 0$
$ \Rightarrow ({x^3} - {x^2} - 6x - {x^2} + x + 6) = 0$
$ \Rightarrow {x^3} - 2{x^2} - 5x + 6 = 0$

We can find the equation (polynomial) by using zero factor property in reverse.

Note:
The common mistake students can do in this problem is the sign of the roots. So make sure you do the individual steps to avoid this mistake.