Question

How do you divide $\left( 3{{x}^{2}}-29x+56 \right)\div \left( x-7 \right)?$
 (a) Using the long division
(b) Using trigonometric identities
(c) Using linear formulas
(d) None of these

Answer 1

Hint: Here in the given problem we are trying to find the solution with the method of long divisions by polynomials where a polynomial is divided by another polynomial like normal division method. If we divide a by b and we get q as quotient and r as a remainder we can write the equation as, a = bq + r. Here we multiply the divisor with such a term that gives us the exact term as the highest power in the dividend and then we can try to proceed in the same way. We multiply the divisor with such a factor so that we cancel out the highest power of the variable in it.

Complete step by step solution:
We have our dividend as, $\left( 3{{x}^{2}}-29x+56 \right)$ and we have our divisor as, $\left( x-7 \right)$
Now, we will try to find the solution using the long division method of polynomials.
$x-7\overline{\left){\begin{align}
  & 3{{x}^{2}}-29x+56| \\
 & \dfrac{3{{x}^{2}}-21x}{\dfrac{\begin{matrix}
   -8x & +56 \\
   -8x & +56 \\
\end{matrix}}{0}} \\
\end{align}}\right.}3x-8$
So, we have started with multiplying the divisor with 3x first and then after simplifying them, we multiply them with – 8 we are getting the solution.
When we divide a by b and we get q as quotient and r as a remainder we can write the equation as, a = bq + r,
Thus, this equation can be written as, $\left( 3{{x}^{2}}-29x+56 \right)=\left( 3x-8 \right)\left( x-7 \right)+0$
Hence, the solution is, (a) Using the long division.

Note: Long division method is the way of dividing the polynomials just like we do it for the numbers. It is the best way to find the quotient from the dividend and divisor as polynomials. There also can exist missing terms in the case of dividing polynomials, in that case we replace the coefficients with 0 and get on with it.