Is (x – 5) a root of x^4 – 3x^3 + x – 4?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Denoting x^4 – 3x^3 + x – 4 by f(x), according to the remainder theorem if x – 5 is a root of x^4 – 3x^3 + x – 4, then f (5) = 0

Now, f (5)

=> 5^4 – 3*5^3 + 5 – 4

=> 25*25 – 3*125 + 5 – 4

=> 625 – 375 + 5 – 4

=> 250 + 5 - 4

=> 251.

As f (5) is not equal to zero, (x – 5) is not a root of x^4 – 3x^3 + x – 4.

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neela | High School Teacher | (Level 3) Valedictorian

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To examine if (x – 5) a factor of f(x) = x^4 – 3x^3 + x – 4, or if x= 5 is a root of f(x) = 0.

If x=5 is a root of f(x) = x^4 – 3x^3 + x – 4, then f(5) = 0.

By remainder theorem if the remainder of f(x) divided by (x-a)  is  f(a).

So the remainder f(a) = 0, if x-a  is a factor of f(x).

So the remainder of (x^4-3x^3+x-4)/(x-5) is 5^4 - 3*5^3+5-4.

=  5^4 - 3*5^3+5-4 = 625-3*125+5 -4 = 251.

Therefore (x-5) does not divide x^4-3x^3+x-4 .

So f(5) = 251 and f(5) is not equal to zero.

Therefore x-5 is not a factor of x^4 – 3x^3 + x – 4. And x= 5 is not a root of x^4 – 3x^3 + x – 4 = 0.

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