Determine the degree of each of the following polynomials.
Question:

Determine the degree of each of the following polynomials.

(i) $\frac{4 x-5 x^{2}+6 x^{3}}{2 x}$

(ii) $y^{2}\left(y-y^{3}\right)$

(iii) $(3 x-2)\left(2 x^{3}+3 x^{2}\right)$

(iv) $-\frac{1}{2} x+3$

(v) $-8$

(vi) $x^{-2}\left(x^{4}+x^{2}\right)$

Solution:

(i) $\frac{4 x-5 x^{2}+6 x^{3}}{2 x}=\frac{4 x}{2 x}-\frac{5 x^{2}}{2 x}+\frac{6 x^{3}}{2 x}=2-\frac{5}{2} x+3 x^{2}$

Here, the highest power of $x$ is 2 . So, the degree of the polynomial is 2 .

(ii) $y^{2}\left(y-y^{3}\right)=y^{3}-y^{5}$

Here, the highest power of $y$ is 5 . So, the degree of the polynomial is 5 .

(iii) $(3 x-2)\left(2 x^{3}+3 x^{2}\right)=6 x^{4}+9 x^{3}-4 x^{3}-6 x^{2}=6 x^{4}+5 x^{3}-6 x^{2}$

Here, the highest power of $x$ is 4 . So, the degree of the polynomial is 4 .

(iv) $-\frac{1}{2} x+3$

Here, the highest power of $x$ is $1 .$ So, the degree of the polynomial is 1 .

(v) – 8
–8 is a constant polynomial. So, the degree of the polynomial is 0.

(vi) $x^{-2}\left(x^{4}+x^{2}\right)=x^{2}+x^{0}=x^{2}+1$

Here, the highest power of $x$ is 2 . So, the degree of the polynomial is 2 .