Question

A trader bought \[2{x^2} - x + 2\] TV sets for Rs.\[(8{x^4} + 7x - 6)\]. Find the price of each TV set.

Answer 1

Hint: We solve for the price of each TV set using the unitary method by dividing the total price of TV sets by the number of TV sets bought by the trader. Then we divide the polynomials using the long division method.
* Unitary method helps us to find the value of single unit by dividing the value of multiple units by the number of units.
* Long division method: when dividing \[a{x^n} + b{x^{n - 1}} + ....c\] by \[px + q\] we perform as

\[px + q)\overline {a{x^n} + b{x^{n - 1}} + ....c} ((a/p){x^{n - 1}} + ...\]
            \[\underline { - a{x^n} + (qa/p){x^{n - 1}}} \]
                          \[0.{x^n} + (b - qa/p){x^{n - 1}}\]

Here we multiply the divisor with such a term that gives us the exact same term as the highest power in the dividend and then proceed in the same way. We multiply the divisor with such a factor so we cancel out the highest power of variable in it.

Complete step by step answer:
We are given that trader bought \[2{x^2} - x + 2\]TV sets for Rs.\[(8{x^4} + 7x - 6)\]
Number of TV sets the trader bought \[ = 2{x^2} - x + 2\]
Price of \[2{x^2} - x + 2\] TV sets bought by the trader \[ = (8{x^4} + 7x - 6)\]
Then we can find price of 1 TV set bought by the trader using unitary method. We divide the total price of TV sets by number of TV sets to find price of one TV set.
\[ \Rightarrow \] Price of one TV set \[ = \dfrac{{8{x^4} + 7x - 6}}{{2{x^2} - x + 2}}\]
We can write \[8{x^4} + 7x - 6\] is divided by \[2{x^2} - x + 2\]
Now we divide the polynomial in the numerator by polynomial in the denominator using the long division method.
Here the dividend is \[8{x^4} + 7x - 6\] and the divisor is \[2{x^2} - x + 2\].
Multiply the divisor with a factor that cancels out the highest power of the equation in the dividend and after the highest power term becomes zero we do the same to the next highest power term.

\[2{x^2} - x + 2)\overline {8{x^4} + 7x - 6} (4{x^2} + 2x - 3\]
                   \[\underline { - 8{x^4} - 4{x^3} + 8{x^2}} \]
                           \[4{x^3} - 8{x^2} + 7x - 6\]
                          \[\underline { - 4{x^3} - 2{x^2} + 4x} \]
                                          \[ - 6{x^2} + 3x - 6\]
                                          \[\underline { - 6{x^2} + 3x - 6} \]
                                                   \[0\]

So, by long division method
\[ \Rightarrow \dfrac{{8{x^4} + 7x - 6}}{{2{x^2} - x + 2}} = 4{x^2} + 2x - 3\]

Therefore, price of each TV set is Rs.\[4{x^2} + 2x - 3\].

Note:
Students are likely to make mistakes while performing the long division method, always keep in mind that the sign needs to be changed from negative to positive and vice versa inside the division when we are solving for the next value to be divided by the dividend.
Students should know the process of long division which is just like basic division just involving polynomial equations and the sign change which is the most important part.