a.Show in factor theorem that x+1 is factor of x^25+1 b.show x intercepts in polynomialx^4-1
a) You need to remember the fact that the linear binomial x-p denotes a factor for a polynomial `P(x), ` then p cancels the polynomial such that `P(p)=0` .
Hence, if you need to verify that `x + 1 = x - (-1)` is a factor for binomial `x^25 + 1` , then you need to check if -1 cancels the binomial such that:
`(-1)^25 + 1 = -1 + 1 = 0`
Notice that x = -1 cancels the binomial `x^25 + 1` , hence x + 1 denotes a factor for `x^25 + 1` .
b) You need to remember that the graph of function intercepts x axis if y=0, hence you need to find the zeroes of polynomial `x^4-1 ` to find x intercepts.
Hence, you should solve `x^4 - 1 = 0` .
You should substitute for `x^4 - 1` such that:
`(x^2- 1)(x^2 + 1) = 0 =gt x^2 - 1 = 0 =gt x_(1,2) = +-1`
`x^2 + 1 = 0 =gt x_(3,4) = +-sqrt(-1) = +- i` (complex number theory)
Hence, the graph intercepts x axis only at points (-1,0) and (1,0) because x intercepts need to be real numbers.
Hence, the binomial `x + 1` is a factor for`x^25+1` and the graph of `y=x^4-1` intercepts x axis at (-1,0) and (1,0).