please tell me not only answers,but also explanations a. Factor polynomial P(X) = x^5 - 256x into linear and irreducible quadratic factor with real coefficients. b. find the polynomial with real coefficients of the smallest possible degree where \'i\' and \'1+i\' are zeros and the coefficient of the highest power is 1 Solution a. p(x)=x*(x^4-256)=x*(x^2+16)(x^2-16)=x*(x^2+16)*(x+4)*(x-4) b. Since i and (i+1) are zeros so -i and -i+1 will also be zeros. Polynomial will be P(x) = (x-i)(x+i)(x-i-1)(x+i-1) P(x) = (x-i)(x+i)(x-1-i)(x-1+i) P(x) = (x^2 - i^2)((x -1)^2 -i^2) P(x) = (x^2 +1)((x -1)^2 +1) P(x) = (x^2 +1)(x^2 -2x+1 +1) P(x) = (x^2 +1)(x^2 -2x+2) P(x) = x^4 - 2x^3 + 3x^2 -2x + 2 Answer: P(x) = x^4 - 2x^3 + 3x^2 -2x + 2.

1. please tell me not only answers,but also explanations
a. Factor polynomial P(X) = x^5 - 256x into linear and irreducible quadratic factor with real
coefficients.
b. find the polynomial with real coefficients of the smallest possible degree where 'i' and '1+i'
are zeros and the coefficient of the highest power is 1
Solution
a. p(x)=x*(x^4-256)=x*(x^2+16)(x^2-16)=x*(x^2+16)*(x+4)*(x-4)
b. Since i and (i+1) are zeros so -i and -i+1 will also be zeros.
Polynomial will be
P(x) = (x-i)(x+i)(x-i-1)(x+i-1)
P(x) = (x-i)(x+i)(x-1-i)(x-1+i)
P(x) = (x^2 - i^2)((x -1)^2 -i^2)
P(x) = (x^2 +1)((x -1)^2 +1)
P(x) = (x^2 +1)(x^2 -2x+1 +1)
P(x) = (x^2 +1)(x^2 -2x+2)
P(x) = x^4 - 2x^3 + 3x^2 -2x + 2
Answer: P(x) = x^4 - 2x^3 + 3x^2 -2x + 2