Question

What is the standard form of a polynomial ${\left( {2x - 6} \right)^2}$?

Answer 1

Hint: A polynomial is defined as an expression which is used of constants and variables and exponents .And combined with the mathematical operations. A polynomial within a standard form is written as the highest degree is the first term of the polynomial. The second term is the second highest degree of the polynomial .This is the way to write the standard form of a polynomial.
Formula used:
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

Complete step-by-step solution:
The given polynomial is ${\left( {2x - 6} \right)^2}$ …………………………..$\left( 1 \right)$
We know that the formula for ${\left( {a - b} \right)^2}$;
Expand the formula, we have,
${\left( {a - b} \right)^2} = \left( {a - b} \right)\left( {a - b} \right)$
Multiply inside the bracket,
$ = {a^2} - ba - ab + {b^2}$
By commutative property, $ba = ab$
Therefore, apply the commutative property, we get,
$ = {a^2} - ab - ab + {b^2}$
Add the values, we have
${\left( {a - b} \right)^2}$$ = {a^2} - 2ab + {b^2}$ ……………………………….$\left( 2 \right)$
$\left( { - ab + - ab = - 2ab} \right)$
Compare the equation $\left( 1 \right)$ and $\left( 2 \right)$,
We have,
$a = 2x$ And$b = 6$;
Substitute these values in equation $\left( 2 \right)$, we get the equation,
${\left( {2x - 6} \right)^2} = {\left( {2x} \right)^2} - 2\left( {2x} \right)\left( 6 \right) + {6^2}$
Squaring and multiplying the values we get,
${\left( {2x - 6} \right)^2} = 4{x^2} - \left( {4x} \right)\left( 6 \right) + 36$
${\left( {2x - 6} \right)^2} = 4{x^2} - 24x + 36$
Therefore the standard form of a given polynomial is
${\left( {2x - 6} \right)^2} = 4{x^2} - 24x + 36$ ……………………………………………………………$\left( 3 \right)$
Otherwise we write this polynomial as,
Simplify the equation $\left( 3 \right)$, dividing the whole equation by four ,we have
${\left( {2x - 6} \right)^2} = {x^2} - 6x + 9$ …………………………………………………………………$\left( 4 \right)$
Because, in equation three the number four is common. All the numbers are multiples of four. So the equation $\left( 3 \right)$ is simplified as equation four. Both the equation $\left( 3 \right)$ and equation $\left( 4 \right)$ have the same and equal value.
Hence the standard form of a given polynomial is
${\left( {2x - 6} \right)^2} = 4{x^2} - 24x + 36$
(Or)
${\left( {2x - 6} \right)^2} = {x^2} - 6x + 9$

Note: In Mathematics, a polynomial is an expression consisting of variables also called intermediates. Polynomial comes from poly means many and nominal (in this we have constant, variables and exponents). There are special names for polynomials with one; two and three terms are monomial, binomial, trinomial. The standard form for writing a polynomial is to write highest degree first and next term with second highest degree and so on.