Question

How do you simplify $\left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$ ?

Answer 1

Hint: For finding the simplification of the polynomial with three variables with maximum two degrees for one variable as given question $\left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$ , we will follow two steps as:
First step: Combining equal like terms and
Second step: arrange the terms in descending order of power.

Complete step by step solution:
Since, $\left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$ is the given question that is a polynomial with three variables i.e. $x,$ $y$ and $z$ . It has two degree for variable $x$ that is the highest degree for any variable among the three variables in the given question.
The given polynomial equation:
$\Rightarrow \left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$
Since, the variables are not in sequence. Here we can write the variables of the equation in a sequence like $xyz$ or $yzx$ or $zxy$ as:
$\Rightarrow \left( -7{{x}^{2}}yz+3{{x}^{2}}yz-{{x}^{2}}yz-4y \right)$
Let us find the equal like terms from the above equation that can be added or subtracted from each other. The above equation is:
$\Rightarrow \left( -7{{x}^{2}}yz+3{{x}^{2}}yz-{{x}^{2}}yz-4y \right)$
By looking the equation, we can say that $-7{{x}^{2}}yz$ , $3{{x}^{2}}yz$ and $-{{x}^{2}}yz$ are like terms with respect to variables and its degrees.
Now, here we will combine equal like terms as:
$\Rightarrow \left( -7{{x}^{2}}yz+3{{x}^{2}}yz-{{x}^{2}}yz \right)-4y$
$\Rightarrow \left( -7+3-1 \right){{x}^{2}}yz-4y$
$\Rightarrow \left( -5{{x}^{2}}yz \right)-4y$
After removing bracket, we can write the above polynomial equation as:
$\Rightarrow -5{{x}^{2}}yz-4y$
Since, the simplified value of the given polynomial is already organized; there is no need to rearrange it.
Hence the simplified value of the given polynomial $\left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$ is $\left( -5{{x}^{2}}yz-4y \right)$ .

Note: Here, we will use of an alternate method of the solution of the given question.
Since, the given polynomial question is not a FOIL problem. So, we don’t need to expand it. It is already in expanded form as
$\Rightarrow \left( -7{{x}^{2}}yz+3yz{{x}^{2}}-z{{x}^{2}}y-4y \right)$ .
From observation we can see that $y$ is the common factor for all the terms of polynomial. So, we can write it outside the bracket as:
$\Rightarrow \left( -7{{x}^{2}}z+3z{{x}^{2}}-z{{x}^{2}}-4 \right)y$
Here, we observe that the terms in the bracket has also some common element like ${{x}^{2}}z$ , so we can write the previous polynomial equation after taking outside this factor as:
$\Rightarrow \left[ \left( -7+3-1 \right){{x}^{2}}y-4 \right]y$
Now, we will calculate the internal values that are in the small bracket and we will get $-5$ as resultant. So, the above equation will be as:
$\Rightarrow \left[ -5{{x}^{2}}z-4 \right]y$
 Here, we will open the bracket again and have the equation as:
$\Rightarrow \left[ -5{{x}^{2}}yz-4y \right]$
This is the simplified value of the given polynomial. Hence, the solution is correct.