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a.Show in factor theorem that x+1 is factor of x^25+1 b.show x intercepts in...

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rouche | Student, Undergraduate | eNoter

Posted February 5, 2012 at 3:52 PM via web

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a.Show in factor theorem that x+1 is factor of x^25+1

b.show x intercepts in polynomialx^4-1

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 5, 2012 at 5:42 PM (Answer #1)

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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).

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