Question

How do you write $4{{y}^{3}}-4{{y}^{2}}+3-y$ in standard form?

Answer 1

Hint: The standard form of a polynomial expression in $x$is of the form ${{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n-2}}{{x}^{n-2}}+...+{{a}_{1}}x+{{a}_{0}}.$ That is, the variables always keep an order. The variables are placed in the equation in the decreasing order of their exponents.

Complete step by step answer:
Let us consider the given problem.
We are asked to write the polynomial expression $4{{y}^{3}}-4{{y}^{2}}+3-y$ in the standard form.
We know that standard form of the general polynomial expression of degree $n$ is of the form ${{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n-2}}{{x}^{n-2}}+...+{{a}_{1}}x+{{a}_{0}}.$ In this form, ${{a}_{n}},{{a}_{n-1}},...{{a}_{1}},{{a}_{0}}$ are constant coefficients of the variables having different exponents with ${{a}_{n}}\ne 0.$
Also, remember that ${{x}^{0}}=1.$ Therefore, ${{a}_{0}}={{a}_{0}}{{x}^{0}}.$
Now, we need to keep another fact in mind. The coefficients can be negative or positive regardless of the exponents of the variables.
So, the leading coefficient, that is the coefficient of the term ${{y}^{3}},$ is $4.$
The coefficient of the second term ${{y}^{2}}$ is $-4.$
And, as we can see, $3$ is the constant term. We can say that $3$ is the coefficient of the variable with exponent $0,$ that is ${{y}^{0}}.$
And finally, we have the term $-y.$ As we can see, $-1$ is the coefficient of this term. Also, remember that the exponent of this term is $1.$
In the descending order of the exponents, the variables are ${{y}^{3}},{{y}^{2}},{{y}^{1}}=y,$ and ${{y}^{0}}=1.$
So, we only need to exchange the positions of $3$ and $-y$ with each other to get the standard form.
Hence the standard form of the given equation is $4{{y}^{3}}-4{{y}^{2}}-y+3.$

Note:
The given polynomial is a polynomial of degree $3$ with constant coefficients. We should always remember that in the standard form, the variables are arranged in the descending order of the exponents. If we write the polynomial in the ascending order of the exponents, then it is not the standard form of the polynomial.