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Question: Find the value of \({{x}^{4}}-{{x}^{3}}+{{x}^{2}}+3x-5\) if \(x=2+3i\)...

Find the value of x4−x3+x2+3x−5{{x}^{4}}-{{x}^{3}}+{{x}^{2}}+3x-5 if x=2+3ix=2+3i

Explanation

Solution

Hint: In this question, we are given the value of x as a complex number and we have to find the value of the given expression of x. Therefore, we should use the formula to find the powers of a complex number and evaluate each term with the given value of x and then take the sum of the terms to get the required answer.

Complete step-by-step answer:

The given expression is x4−x3+x2+3x−5{{x}^{4}}-{{x}^{3}}+{{x}^{2}}+3x-5. We know that the value of i in complex numbers is given by i=−1i=\sqrt{-1} .
Also, we know that the square of a sum of two number which may be complex is given by
(a+b)2=a2+2ab+b2.......................(1.1){{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}.......................(1.1)
Therefore, using (1.1) , we can evaluate x2{{x}^{2}} as
x2=(2+3i)2=22+2×2×3i+(3i)2 =4+12i+9×i2=4+12i+9×−1=−5+12i.............(1.2) \begin{aligned} & {{x}^{2}}={{\left( 2+3i \right)}^{2}}={{2}^{2}}+2\times 2\times 3i+{{\left( 3i \right)}^{2}} \\\ & =4+12i+9\times {{i}^{2}}=4+12i+9\times -1=-5+12i.............(1.2) \\\ \end{aligned}
And to find x4{{x}^{4}} , we can take the square of x2{{x}^{2}} , thus, using equation (1.2), we get
x4=(x2)2=(−5+12i)2=(−5)2+2×−5×12i+(12i)2 =25−120i+122i2=25−120i+144×−1=−119−120i............(1.3) \begin{aligned} & {{x}^{4}}={{\left( {{x}^{2}} \right)}^{2}}={{\left( -5+12i \right)}^{2}}={{\left( -5 \right)}^{2}}+2\times -5\times 12i+{{\left( 12i \right)}^{2}} \\\ & =25-120i+{{12}^{2}}{{i}^{2}}=25-120i+144\times -1=-119-120i............(1.3) \\\ \end{aligned}
To find x3{{x}^{3}} , we can multiply x2{{x}^{2}} from (1.2) with x to obtain
x3=x2×x=(−5+12i)(2+3i)=−10−15i+24i+36i2 =−10+9i−36=−46+9i.........................(1.4) \begin{aligned} & {{x}^{3}}={{x}^{2}}\times x=\left( -5+12i \right)\left( 2+3i \right)=-10-15i+24i+36{{i}^{2}} \\\ & =-10+9i-36=-46+9i.........................(1.4) \\\ \end{aligned}
Therefore, using (1.2), (1.3) and (1.4), we can find the value of the given expression as
x4−x3+x2+3x−5=(−119−120i)−(−46+9i)+(−5+12i)+3×(2+3i)−5 =(−119+46−5+6−5)+i(−120−9+12+9) =−77−108i \begin{aligned} & {{x}^{4}}-{{x}^{3}}+{{x}^{2}}+3x-5=\left( -119-120i \right)-\left( -46+9i \right)+\left( -5+12i \right)+3\times \left( 2+3i \right)-5 \\\ & =\left( -119+46-5+6-5 \right)+i\left( -120-9+12+9 \right) \\\ & =-77-108i \\\ \end{aligned}
Thus, the required answer to this question is -77-108i.

Note: We could also have tried to factorise the given expression into a product of terms which are linear in x. Then, we could have put the value of x to obtain the answer. However, as the expression is of fourth power in x, factorizing it will be very time consuming and difficult. Therefore, directly evaluating the expression by putting the value of x is a simple and straightforward approach to these types of problems.