Question

find the value of x^4+9x^3+35x^2-x+164 ;if x=-5+4i​

Answer 1

Answer:

${x}^{4} + 9 {x}^{3} + 35 {x}^{2} - x + 164 = 0 \\ \\ if \: x = - 5 + 4i \\ \:$

Step-by-step explanation:

We know that

${(x - y)}^{2} = {x}^{2} + {y}^{2} - 2xy$

as according to the question

$x = - 5 + 4i \\ \\ {( - 5 + 4i)}^{2} = 25 - 16 - 40i \\ \\ = 9 - 40i...eq1 \\ \\ \because \: {i}^{2} = - 1 \\ \\ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3 {x}^{2}y + 3x {y}^{2} \\ \\ {( 4i - 5)}^{3} = {(4i)}^{3} - {(5)}^{3} - 3 {(4i)}^{2}(5) + 3(4i) {( - 5)}^{2} \\ \\ = - 64i - 125 + 240 + 300i \\ \\ = 115 + 236i...eq2$

$= (9 - 40i)(9 - 40i) \\ \\ = {(9 - 40i)}^{2} \\ \\ = 81 - 1600 - 720i \\ \\ = - 1519 - 720i \: ...eq3$

put the values from eq1,2 and 3 in

${x}^{4} + 9 {x}^{3} + 35 {x}^{2} - x + 164 \\ \\ = - 1519 - 720i + 9(115 + 236i) + 35(9 - 40i) - ( - 5 + 4i) + 164 \\ \\ = - 1519 + 1035 + 315 + 5 + 164 + i( - 720 + 2124 - 1400 - 4) \\ \\ = 0 + i(0) \\ \\ = 0 \\ \\ {x}^{4} + 9 {x}^{3} + 35 {x}^{2} - x + 164 = 0 \\ \\ if \: x = - 5 + 4i$