Question

How do you find the degree of $\left( x+4 \right){{\left( x+1 \right)}^{2}}{{\left( x-1 \right)}^{3}}$

Answer 1

Hint: Degree of any polynomial is the highest power of variable x so in the above polynomial expression, degree will be the highest power of variable x. The above expression is the product of different powers of variable x so we don’t want to fully open each bracket and then multiply them and get the highest power instead we will take out the highest power of x from that bracket and then multiply them.

Complete step by step solution:
The polynomial given in the above problem is as follows:
$\left( x+4 \right){{\left( x+1 \right)}^{2}}{{\left( x-1 \right)}^{3}}$
Now, we are going to find the degree of the above polynomial by expanding each of the brackets as follows:
$\Rightarrow \left( x+4 \right)\left( {{x}^{2}}+2x+1 \right)\left( {{x}^{3}}-3{{x}^{2}}+3x-1 \right)$
The above expansion is written by using the following algebraic identities:
$\begin{align}
  & {{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}; \\
 & {{\left( a-b \right)}^{3}}={{a}^{3}}-3{{a}^{2}}b+3a{{b}^{2}}-{{b}^{3}} \\
\end{align}$
Now, we are going to multiply the expanded brackets and get the answer:
$=\left( {{x}^{3}}+2{{x}^{2}}+x+4{{x}^{2}}+8x+4 \right)\left( {{x}^{3}}-3{{x}^{2}}+3x-1 \right)$
Now, there is no need to multiply the two brackets completely instead multiply the variables of x with highest power corresponding to each bracket and we get,
As you can see that the highest power of first bracket is 3 and highest power of x in the second bracket is 3 so multiplying the variable x of these powers we get,
$\Rightarrow {{x}^{3}}\left( {{x}^{3}} \right)$
We know there is a property that if the base is the same and the base is written with the multiplication sign then powers of the same bases are added up.
$\Rightarrow {{a}^{n}}\times {{a}^{m}}={{a}^{n+m}}$
Applying the above property into the above variables power we get,
$\Rightarrow {{x}^{3+3}}={{x}^{6}}$

Hence, we have got the highest power (or degree) of the polynomial as 6.

Note: In the above problem, instead of opening each of the brackets you can take the highest power of x from each bracket and then multiply those variables with highest power and will get the degree of the polynomial.
The polynomial given above as follows:
$\left( x+4 \right){{\left( x+1 \right)}^{2}}{{\left( x-1 \right)}^{3}}$
From the first bracket we will get $x$, from the second bracket we get ${{x}^{2}}$ and from the third bracket we will get ${{x}^{3}}$. Now, multiplying each of these variables x we get,
$\begin{align}
  & \Rightarrow x\times {{x}^{2}}\times {{x}^{3}} \\
 & ={{x}^{1+2+3}}={{x}^{6}} \\
\end{align}$
This is the alternate approach to the above problem.