The degree of the polynomial
$f(x)=(x^{2}+1)(x^{3}+2)(x+5)$ is :-

Question

The degree of the polynomial 
$f(x)=(x^{2}+1)(x^{3}+2)(x+5)$ is :-

Accepted Answer

The correct option is (3):
To find the degree of the polynomial $f(x)=(x^{2}+1)(x^{3}+2)(x+5)$, we need to understand what the degree of a polynomial is. The degree of a polynomial is the highest power of the variable in the polynomial.

When we multiply polynomials, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials being multiplied.

Let's identify the degree of each factor in the given polynomial:1.  The first factor is $(x^{2}+1)$. The highest power of $x$ in this factor is $2$. So, the degree of $(x^{2}+1)$ is $2$.
2.  The second factor is $(x^{3}+2)$. The highest power of $x$ in this factor is $3$. So, the degree of $(x^{3}+2)$ is $3$.
3.  The third factor is $(x+5)$. The highest power of $x$ in this factor is $1$ (since $x = x^1$). So, the degree of $(x+5)$ is $1$.Now, to find the degree of the product polynomial $f(x)$, we add the degrees of these individual factors:
Degree of $f(x)$ = Degree of $(x^{2}+1)$ + Degree of $(x^{3}+2)$ + Degree of $(x+5)$
Degree of $f(x)$ = $2 + 3 + 1 = 6$.

Thus, the degree of the polynomial $f(x)=(x^{2}+1)(x^{3}+2)(x+5)$ is $6$.

The incorrect option is (1):
This option suggests the degree is $1$. This would be the case if the polynomial was linear, which it is not. A common mistake here might be to consider only the degree of the last term if the polynomial was not in factored form, or to confuse it with the constant term.

The incorrect option is (2):
This option suggests the degree is $7$. This might result from an error in adding the degrees, such as incorrectly counting one of the degrees or adding an extra power. For example, if one mistakenly considered the degree of $(x+5)$ as $2$ instead of $1$, the sum would be $2+3+2 = 7$.

The incorrect option is (4):
This option suggests the degree is $3$. This might happen if one only considers the highest degree from one of the factors, such as $x^3+2$, and overlooks the contribution of the other factors to the overall degree of the product.