Question

What is the leading term, leading coefficient and degree of this polynomial $4y+18y^{2} +8-10y^{4} ?$

Answer 1

Hint: For this type of questions, we need to remember the concept of leading term, leading coefficient and degree of the polynomial with their definitions. The terms with highest degree of variable will be the leading term, the coefficient of the term with highest degree of variable will be the leading coefficient and degree would be the highest power of a variable in the polynomial.

Complete step by step solution:
The leading term of a polynomial is defined as the term with the highest degree of variable. So, if we consider a polynomial in variable $x$ as an example, we can write it as $ -4x^{3} +7x-9$.
In the above polynomial, the leading term will be $-4x^{3} $ as $x$ has the highest degree.
We know that the polynomial’s leading coefficient is defined as the coefficient of the leading term.
So, in the above example, we get the leading coefficient is $-4$.
We can define the degree of a polynomial as the highest power of a variable in a polynomial expression. In our example, the degree of the polynomial is $3$.
The objective of the given problem is to find the leading term, leading coefficient and degree of this polynomial $4y+18y^{2} +8-10y^{4}$.
Let us assume $f(y)=4y+18y^{2} +8-10y^{4}$.
Rearranging the terms in the given polynomial $f(y)$ becomes, $f(y)=-10y^{4}+18y^{2}+4y+8$.
Now, we are finding the leading term of the $f(y)=-10y^{4}+18y^{2}+4y+8$.
By recalling, the term with highest degree is called the leading term.
So, the leading term in $f(y)=-10y^{4}+18y^{2}+4y+8$. is $-10y^{4}$.
Leading coefficient is simply the number which multiplies the leading term.
In the given polynomial, we have a leading term as $-10y^{4}$ and the number which multiplies the leading term is $-10$.
Degree of a polynomial is the highest power of a variable, in a given polynomial.
In the given expression we have $4,2,1$ as the exponents.
The degree of $f(y)=-10y^{4} +18y^{2} +4y+8$ is 4.

Note: Firstly, given polynomials should be arranged in sequence for easy calculations, most of the students don’t do that and also check whether the leading coefficient is positive or negative. If we have two variables in a single term of a polynomial, then we have to add the exponents of each variable and then find the degree of that term. Only after that we should look for the highest power of variable in the terms to find the degree.