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Write the degree of each of the following polynomials:
(i) $2x^3+5x^2-7$
(ii) $5x^2-3x+2$
(iii) $2x+x^2-8$
(iv) $\frac{1}{2}y^7-12y^6+48y^5-10$
(v) $3x^3+1$
(vi) $5$
(vii) $20x^3+12x^2y^2-10y^2+20$
Given:
The given polynomials are:
(i) $2x^3+5x^2-7$
(ii) $5x^2-3x+2$
(iii) $2x+x^2-8$
(iv) $\frac{1}{2}y^7-12y^6+48y^5-10$
(v) $3x^3+1$
(vi) $5$
(vii) $20x^3+12x^2y^2-10y^2+20$
To do:
We have to find the degree of each of the given polynomials.
Solution:
Degree of a polynomial:
The degree of a polynomial is the highest or the greatest power of a variable in the polynomial expression.
To find the degree, identify the exponents on the variables in each term, and add them together to find the degree of each term.
(i) The given polynomial is $2x^3+5x^2-7$
The variable in the given polynomial is $x$.
Here,
The power of $x$ in $2x^3$ is $3$.
Therefore,
The degree of the given polynomial is $3$.
(ii) The given polynomial is $5x^2-3x+2$
The variable in the given polynomial is $x$.
Here,
The power of $x$ in $5x^2$ is $2$.
Therefore,
The degree of the given polynomial is $2$.
(iii) The given polynomial is $2x+x^2-8$
The variable in the given polynomial is $x$.
Here,
The power of $x$ in $x^2$ is $2$.
Therefore,
The degree of the given polynomial is $2$.
(iv) The given polynomial is $\frac{1}{2}y^7-12y^6+48y^5-10$
The variable in the given polynomial is $y$.
Here,
The power of $y$ in $\frac{1}{2}y^7$ is $7$.
Therefore,
The degree of the given polynomial is $7$.
(v) The given polynomial is $3x^3+1$
The variable in the given polynomial is $x$.
Here,
The power of $x$ in $3x^3$ is $3$.
Therefore,
The degree of the given polynomial is $3$.
(vi) The given polynomial is $5$
A polynomial having its highest degree zero is called a constant polynomial. The degree of a constant polynomial is $0$.
There is no variable in the given polynomial.
$5$ is a constant number.
Therefore,
The degree of the given polynomial is $0$.
(vii) The given polynomial is $20x^3+12x^2y^2-10y^2+20$
The variables in the given polynomial are $x$ and $y$.
Here,
The degree of the term $20x^3$ is $3$.
The degree of the term $12x^2y^2$ is $2+2=4$.
The degree of the term $-10y^2$ is $2$.
The degree of the term $20$ is $0$.
Therefore,
The degree of the given polynomial is $4$.
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