Question

Find zeros of polynomial \[p(x) = {\left( {x - 2} \right)^2} - {(x + 2)^2}\]

Answer 1

Hint: There are two things that you must be able to do when simplifying algebraic expressions. The first is to be able to use the distributive property. The second math concept that you must understand is how to combine like terms.

Complete step-by-step answer:
\[P\left( x \right) = {\left( {x - 2} \right)^2} - {\left( {x + 2} \right)^2}\]
The product of the sum and difference of the same two terms equals the difference between the squares of those terms. Also shown as:
The difference of Two Squares: \[{a^2}-{\text{ }}{b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
Here we will assume $a = x - 2$ and $b = x + 2$
This is in the form of \[{a^2}-{\text{ }}{b^2}\]and by the formula, we get
\[ \Rightarrow P\left( x \right) = \left( {x - 2 + x + 2} \right)\left( {x - 2 - \left( {x + 2} \right)} \right)\]
In this expression, we can use the distributive property to get rid of the second set of parentheses. The next step in simplifying is to look for terms and combine them.
$ \Rightarrow P\left( x \right) = 2x\left( {x - 2 - x - 2} \right)$
\[ \Rightarrow P\left( x \right) = 2x\left( { - 4} \right)\]
\[ \Rightarrow P\left( x \right) = - 8x\]
For finding the zeros of polynomial \[p(x) = {\left( {x - 2} \right)^2} - {(x + 2)^2}\], we will put $P(x) = 0$
$ \Rightarrow P(x) = 0 = - 8x$
A useful rule is the denominator-numerator rule which states that the denominator and numerator may be multiplied by the same quantity without changing the value of the fraction.
$ \Rightarrow x = 0$

Hence, \[x = 0\;\] is the zero of the polynomial P(x)

Note: The properties of real numbers apply to algebraic expressions, because variables are simply representations of unknown real numbers.
Combine like terms, or terms with the same variable part, to simplify expressions.
Use the distributive property when multiplying grouped algebraic expression.
It is a best practice to apply the distributive property only when the expression within the grouping is completely simplified.
After applying the distributive property, eliminate the parentheses and then combine any like terms.
Always use the order of operations when simplifying.