Question

Subtract $4pq-5{{q}^{2}}-3{{p}^{2}}$ from $5{{p}^{2}}+3{{q}^{2}}-pq$ \[\]

Answer 1

Hint: We denote the polynomials $4pq-5{{q}^{2}}-3{{p}^{2}}$ as ${{f}_{1}}$ and $5{{p}^{2}}+3{{q}^{2}}-pq$ as ${{f}_{2}}$. We convert the subtraction to addition ${{f}_{1}}-{{f}_{2}}={{f}_{1}}+\left( -1 \right){{f}_{2}}$. We put the polynomials and arrange them like terms in decreasing order of $p$. We take the variable(s) common and subtract the coefficients to get the result.

Complete step by step answer:
We know that in a polynomial the parts separated by a + or $-$ sign are called terms.
The two given polynomials are $4pq-5{{q}^{2}}-3{{p}^{2}}$ and $5{{p}^{2}}+3{{q}^{2}}-pq$. There are two variables in both the polynomials and they are the same $p,q$. We see that both the given polynomials are of degree 2 because all the terms in both the polynomials are 2. We denote $4pq-5{{q}^{2}}-3{{p}^{2}}$ as $5{p^2} + 3{q^2} - pq$ as ${{f}_{2}}$.\[\]
We are asked to subtract ${{f}_{1}}$ from ${{f}_{2}}$ which means finding ${{f}_{2}}-{{f}_{1}}$. We know that the terms with same variables with same exponents or variables with same exponents are called like terms. We see that both terms there are coefficients of ${{p}^{2}}$, coefficients of ${{q}^{2}}$ and coefficients of $pq$. They are like terms.
We can write ${{f}_{2}}-{{f}_{1}}$ as ${{f}_{2}}+\left( -1 \right){{f}_{1}}$ and proceed,
\[\begin{align}
  & {{f}_{2}}-{{f}_{1}}={{f}_{2}}+\left( -1 \right){{f}_{1}} \\
 & =5{{p}^{2}}+3{{q}^{2}}-pq+\left( -1 \right)\left( 4pq-5{{q}^{2}}-3{{p}^{2}} \right) \\
 & =5{{p}^{2}}+3{{q}^{2}}-pq-4pq+5{{q}^{2}}+3{{p}^{2}} \\
\end{align}\]
Let us arrange in a way that the like terms are close to each other and arrange them in the decreasing order of $p$. We have,
\[\begin{align}
  & =5{{p}^{2}}+3{{q}^{2}}-pq-4pq+5{{q}^{2}}+3{{p}^{2}} \\
 & =5{{p}^{2}}+3{{p}^{2}}-pq-4pq+3{{q}^{2}}+5{{q}^{2}} \\
\end{align}\]
Let us take the variable or product of variables common and proceed
\[\begin{align}
  & =5{{p}^{2}}+3{{p}^{2}}-pq-4pq+3{{q}^{2}}+5{{q}^{2}} \\
 & ={{p}^{2}}\left( 5+3 \right)-pq\left( 1+4 \right)+{{q}^{2}}\left( 3+5 \right) \\
 & =8{{p}^{2}}-5pq+8{{q}^{2}} \\
\end{align}\]
Alternatively we can solve by arranging both the polynomials in decreasing order of $p$. So we have ${{f}_{1}}=4pq-5{{q}^{2}}-3{{p}^{2}}=-3{{p}^{2}}+4pq-5{{q}^{2}}$ and ${{f}_{2}}=5{{p}^{2}}+3{{q}^{2}}-pq=5{{p}^{2}}-pq+3{{q}^{2}}$. We subtract ${{f}_{1}}$ from ${{f}_{2}}$ by writing the like terms in column
\[\begin{matrix}
   {} & 5{{p}^{2}}-pq+3{{q}^{2}} & {} \\
   - & \underline{-3{{p}^{2}}+4pq-5{{q}^{2}}} & {} \\
   {} & 8{{p}^{2}}-5pq+8{{q}^{2}} & {} \\
\end{matrix}\]

Note:
We can also solve with decreasing order of $q$. The given two polynomials are in two variables $p,q$ which are called binomials. The degree of the polynomials is 2 . The polynomials of degree 2 are called quadratic polynomials. We can multiply two polynomials with distributive property and divide by long division method.