Hrishi98 Posted May 9, 2015 Report Share Posted May 9, 2015 . [7 marks]Consider the polynomialx^4+ax^3+bx^2+cx+d , where a, b, c, d are real numbersGiven that 1 + i and 1 − 2i are zeros of the above polynomial, find the values of a, b, c and d. I'm having tough time as this is very long in solving. Could anyone help out in doing this? Reply Link to post Share on other sites More sharing options...
Ossih Posted May 9, 2015 Report Share Posted May 9, 2015 . [7 marks]Consider the polynomialx^4+ax^3+bx^2+cx+d , where a, b, c, d are real numbersGiven that 1 + i and 1 − 2i are zeros of the above polynomial, find the values of a, b, c and d. I'm having tough time as this is very long in solving. Could anyone help out in doing this? So if 1+i is a zero, that means 1-i is a zero as well right? so (x-1-i) and (x-1+i) are factorsNow, 1-2i is a zero so 1+2i is a zero as well, so (x-1+2i) and (x-1-2i) are factors So now, the polynomial is (x-1-i)(x-1+i)(x-1+2i)(x-1-2i)One way to expand them is to group them like this --> [(x-1)-i][(x-1)+i] [(x-1)+2i][(x-1)-2i]Now you can expand the first two and last two using (a-b)(a+b) = a^2 - b^2so this is[(x-1)^2 -i^2] [(x-1)^2 - 4i^2]= (x^2 - 2x + 1 + 1)(x^2 - 2x + 1 + 4)= (x^2 - 2x + 2)(x^2 - 2x + 5)now expand this out= x^4 - 2x^3 + 5x^2 - 2x^3 +4x^2 - 10x + 2x^2 -4x + 10= x^4 - 4x^3 + 11x^2 - 4x + 10 Hope this helped 1 Reply Link to post Share on other sites More sharing options...
Hrishi98 Posted May 9, 2015 Author Report Share Posted May 9, 2015 . [7 marks]Consider the polynomialx^4+ax^3+bx^2+cx+d , where a, b, c, d are real numbersGiven that 1 + i and 1 − 2i are zeros of the above polynomial, find the values of a, b, c and d. I'm having tough time as this is very long in solving. Could anyone help out in doing this? So if 1+i is a zero, that means 1-i is a zero as well right? so (x-1-i) and (x-1+i) are factorsNow, 1-2i is a zero so 1+2i is a zero as well, so (x-1+2i) and (x-1-2i) are factors So now, the polynomial is (x-1-i)(x-1+i)(x-1+2i)(x-1-2i)One way to expand them is to group them like this --> [(x-1)-i][(x-1)+i] [(x-1)+2i][(x-1)-2i]Now you can expand the first two and last two using (a-b)(a+b) = a^2 - b^2so this is[(x-1)^2 -i^2] [(x-1)^2 - 4i^2]= (x^2 - 2x + 1 + 1)(x^2 - 2x + 1 + 4)= (x^2 - 2x + 2)(x^2 - 2x + 5)now expand this out= x^4 - 2x^3 + 5x^2 - 2x^3 +4x^2 - 10x + 2x^2 -4x + 10= x^4 - 4x^3 + 11x^2 - 4x + 10 Hope this helped Thanks!!! I made a small mistake in the factors thing. Reply Link to post Share on other sites More sharing options...
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