Question

Write the polynomial in standard form and also write down their degree.
\[\left( x+\dfrac{2}{3} \right)\left( x+\dfrac{3}{4} \right)\]

Answer 1

Hint: The standard form of a polynomial is given as \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+\cdots +{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}\]
(Where coefficients of the variables are constants).The degree of a polynomial is the highest power of the variable in the polynomial.Another important thing is that this above mentioned rule can be applied only when we have a polynomial in its standard form that is when we will multiply both the terms in the expression that is given in the question and bring it into the standard form, then only we can get to the degree of the polynomial.

Complete step-by-step answer:
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is $x^2 − 4x + 7$. An example in three variables is $x^3 + 2xyz^2 − yz + 1$.
We have to find the standard form of the given polynomial and the degree of the polynomial as well.
Now, the standard form of the given polynomial can be evaluated as follows
We know that the standard form of polynomial is given as \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+\cdots +{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}\] (Where coefficients of the variables are constants). So converting the given equation into standard form, we get,
\[\begin{align}
  & =\left( x+\dfrac{2}{3} \right)\left( x+\dfrac{3}{4} \right) \\
 & ={{x}^{2}}+\dfrac{17}{12}x+\dfrac{1}{2} \\
\end{align}\]
Hence, this is the standard form of the given polynomial and the degree of this polynomial is 2.

Note: Students should remember the standard form of polynomial which is given as
\[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+\cdots +{{a}_{2}}{{x}^{2}}+{{a}_{1}}x+{{a}_{0}}\]
(where coefficients of the variables are constants). The most important rule has to remember to solve this question that the degree of a polynomial is the highest power of the variable in the polynomial.