Question

Find the remainder when $ p(x) = {x^3} + 4{x^2} - 3x + 10 $ is divided by $ x + 4 $

Answer 1

Hint: In this question, we are given a function that is divided by an algebraic expression; they both are in terms of x. We know the dividend and the divisor so we will use the division algorithm to solve this question. But the value of the quotient is unknown to us so we have only one equation and three unknown quantities (x, quotient and remainder). So to find the value of the remainder, we will put the value of the divisor zero. On putting the value of the x at which the divisor is 0, we can find the value of the remainder.

Complete step by step solution:
We know that $ a = bq + r $
We have $ a = {x^3} + 4{x^2} - 3x + 10 $ and $ b = x + 4 $
We see that when $ b = 0,\,a = r $
So,
 $
  x + 4 = 0 \\
   \Rightarrow x = - 4 \;
  $
At $ x = - 4 $ , the value of a is –
 $
  p( - 4) = {( - 4)^3} + 4{( - 4)^2} - 3( - 4) + 10 \\
   \Rightarrow p( - 4) = - 64 + 64 + 12 + 10 \\
   \Rightarrow p( - 4) = 22 \;
  $
Now $ a = r $ when $ b = 0 $ , so the remainder is equal to 22.
Hence when $ p(x) = {x^3} + 4{x^2} - 3x + 10 $ is divided by $ x + 4 $ , then the remainder is 22.
So, the correct answer is “22”.

Note: When a polynomial equation of degree n is divided by another polynomial of degree m such that $ m \leqslant n $ , then the degree of the quotient is $ n - m $ and the degree of the remainder is $ m - 1 $ . So the degree of the quotient of the above expression is 2 and the remainder is a constant. We know that for finding the values of “n” unknown quantities, we need “n” number of equations, so in this question, we had to eliminate two of the three unknown quantities to find the value of the third one that’s why we used this approach.