Question

Without actual division prove that \[{x^4} + 2{x^3} - 2{x^2} + 2x - 3\] is exactly divisible by \[{x^2} + 2x - 3\].

Answer 1

Hint:
According to the question, firstly we will make the factors of the equation \[{x^2} + 2x - 3\] and we will equate the factors equal to zero to calculate the value x. Thus, put the values of x separately into the given equation that is \[{x^4} + 2{x^3} - 2{x^2} + 2x - 3\] and prove it equal to zero.

Complete step by step solution:
Let us the assume the polynomial \[p(x) = \]\[{x^4} + 2{x^3} - 2{x^2} + 2x - 3\] and the quadratic equation as \[q(x) = \]\[{x^2} + 2x - 3\] .
As we will firstly calculate the factors of the quadratic equation \[q(x)\] by splitting the middle term method.
Here, \[{x^2} + 2x - 3\]
In this method we will find out two numbers whose sum is 2 and product is \[ - 3\].
So, we are getting with the two numbers which are \[ - 1\] and \[3\]
\[ \Rightarrow {x^2} + 3x - x - 3\]
Taking out common in the pairs of 2 we get,
\[ \Rightarrow x(x + 3) - 1(x + 3)\]
Taking 2 same factors one time we get,
\[ \Rightarrow \left( {x + 3} \right)\left( {x - 1} \right)\]
Here, we will substitute the above equation equal to zero \[\left( {x + 3} \right)\left( {x - 1} \right) = 0\]
Now, we will separate the above two factors to calculate the value of x.
Firstly we will take the factor \[x + 3 = 0\]
Taking 6 on the right side we get,
Therefore, \[x = - 3\]
Secondly we will take the factor \[x - 1 = 0\]
Taking 2 on the right side we get,
Therefore, \[x = 1\]
Hence, the value of \[x = 1, - 3\]
Now, we will substitute the values of x in \[{x^4} + 2{x^3} - 2{x^2} + 2x - 3\] .
Firstly put \[x = - 3\]
\[ \Rightarrow {( - 3)^4} + 2{( - 3)^3} - 2{( - 3)^2} + 2( - 3) - 3\]
Now, we will simplify the above equation
So, we get
\[ \Rightarrow 81 - 54 - 18 - 6 - 3\]
Therefore,
\[ \Rightarrow 81 - 81 = 0\]
Secondly, we will put \[x = 1\]
\[ \Rightarrow {(1)^4} + 2{(1)^3} - 2{(1)^2} + 2(1) - 3\]
Now, we will simplify the above equation
So, we get
\[ \Rightarrow 1 + 2 - 2 + 2 - 3\]
Therefore,
\[ \Rightarrow 5 - 5 = 0\]

As, it is clear that \[{x^4} + 2{x^3} - 2{x^2} + 2x - 3\] is exactly divisible by \[{x^2} + 2x - 3\].
Hence Proved.

Note:
To solve these types of questions, we must remember that we should not use a long division method to solve the given equation. Do not forget to calculate the values of x by equating the quadratic equation equal to zero. As we have to make the polynomial exactly divisible by the other polynomial.