Zero Product Property

The zero product property states that if $a \times b = 0$, then $a=0$ or $b=0$, or both.

When factoring expressions both sides, one must be careful with cancelling the zero (null) solutions.

This can be extended to functions, and is a property to solve equations through factorization. For example, we have

$$x^2-6x+5 = 0 \quad \text{ or }\quad (x-1)(x-5) = 0.$$

Using the zero product property, either $(x-1)=0 \text{ or } (x-5) = 0.$ Thus, $x=1$ or $x=5.$

However, this may not be applied to matrices as two matrices A and B can have a product of 0.

Solve $2(x-2)= 5x(x-2)$.

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The above equation can be rewritten as follows:

$$\begin{aligned} 2(x-2)&= 5x(x-2)\\ 5x(x-2)-2(x-2)&=0\\ (5x-2)(x-2) &= 0 . \end{aligned}$$

Using the zero product property, either $(5x-2)=0$ or $(x-2) = 0.$

Thus, $x=\frac{2}{5}$ or $x=2.$ $_\square$

Solve $4x^2 = 64x$.

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We have

$$\begin{aligned} 4x^2 &= 64x \\ 4x^2 - 64x &= 0 \\ 4(x^2 - 16x) &= 0 \\ 4x(x-16) &= 0. \end{aligned}$$

Using the zero product property, either $x=0$ or $(x-16) = 0.$

Thus, $x=0$ or $x=16.$ $_\square$

Solve $(x-2)^2(x-1) = 2(2x-5)(x-2)$.

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The above equation can be rewritten as follows:

$$\begin{aligned} (x-2)^2(x-1) &= 2(2x-5)(x-2) \\ x^3 -5x^2 +8x -4 &= 4x^2 - 18x +20 \\ (x^3 -5x^2 +8x -4) - (4x^2 - 18x +20) &= 0 \\ x^3 - 9x^2 +26x -24 &= 0 \\ (x-2)(x-3)(x-4) &= 0. \end{aligned}$$

Using the zero product property, either $(x-2)=0$ or $(x-3)=0$ or $(x-4) = 0.$

Thus, $x=2$ or $x=3$ or $x=4.$ $_\square$