Question

How do you find the quotient of $ {x^3} - 43x + 42 $ divided by $ {x^2} + 6x - 7 $ ?

Answer 1

Hint: To find the quotient of $ {x^3} - 43x + 42 $ divided by $ {x^2} + 6x - 7 $ , we will write them as a fraction where $ {x^3} - 43x + 42 $ will be the numerator and $ {x^2} + 6x - 7 $ will be the denominator. We will factorize both the numerator and the denominator and then cancel out the factors that are present in both the numerator and the denominator.

Complete step by step solution:
We have to find the quotient of $ {x^3} - 43x + 42 $ divided by $ {x^2} + 6x - 7 $ , so we write it as –
 $ \dfrac{{{x^3} - 43x + 42}}{{{x^2} + 6x - 7}} $
We see that at $ x = 1 $ , $ {x^3} - 43x + 42 $ and $ {x^2} + 6x - 7 $ are zero, that is, $ {(1)^3} - 43(1) + 42 = - 42 + 42 = 0 $ and $ {(1)^2} + 6(1) - 7 = 7 - 7 = 0 $
So, $ x - 1 $ is one of the factors of both the equations.
We can write $ {x^3} - 43x + 42 $ as $ {x^3} - {x^2} + {x^2} - x - 42x + 42 $
 $
   \Rightarrow {x^3} - 43x + 42 = {x^2}(x - 1) + x(x - 1) - 42(x - 1) \\
   \Rightarrow {x^3} - 43x + 42 = ({x^2} + x - 42)(x - 1) \;
  $
Now,
 $
  {x^2} + x - 42 = {x^2} + 7x - 6x - 42 \\
   \Rightarrow {x^2} + x - 42 = x(x + 7) - 6(x + 7) \\
   \Rightarrow {x^2} + x - 42 = (x - 6)(x + 7) \\
   \Rightarrow {x^3} - 43x + 42 = (x - 6)(x + 7)(x - 1) \;
  $
We can write $ {x^2} + 6x - 7 $ as $ {x^2} - x + 7x - 7 $
 $
   \Rightarrow {x^2} + 6x - 7 = {x^2} - x + 7x - 7 \\
   \Rightarrow {x^2} + 6x - 7 = x(x - 1) + 7(x - 1) \\
   \Rightarrow {x^2} + 6x - 7 = (x - 1)(x + 7) \;
  $
Putting these values in the fraction $ \dfrac{{{x^3} - 43x + 42}}{{{x^2} + 6x - 7}} $ , we get –
 $
  \dfrac{{{x^3} - 43x + 42}}{{{x^2} + 6x - 7}} = \dfrac{{(x - 6)(x + 7)(x - 1)}}{{(x + 7)(x - 1)}} \\
   \Rightarrow \dfrac{{{x^3} - 43x + 42}}{{{x^2} + 6x - 7}} = x - 6 \;
  $
Hence the quotient of $ {x^3} - 43x + 42 $ divided by $ {x^2} + 6x - 7 $ is $ x - 6 $ .
So, the correct answer is “ $ x - 6 $ ”.

Note: The factors of an equation are defined as those values that completely divide the equation without leaving any denominator. We are given two polynomial equations as the unknown variable quantity (x) is raised to a non-negative integer as the power in both the equations. The degree of the first equation is 3 as the highest exponent in the first equation is 3, and the degree of the second equation is 2 as the highest exponent in the second equation is 2. We know that a degree has exactly as many factor as the degree of the equation, that’s why the first equation has 3 factors and the second equation has 2 factors. On canceling out the common factors we got the quotient.