Question

show that a-1 is a factor of f=a^100+a^90+a^80-a^70-a^60-1​

Answer 1

$\huge\mathfrak\blue{Answer:}$

Given:

• We have been given a Polynomial
• $f(a) = {a}^{100} + {a}^{90} + {a}^{80} + {a}^{70} + {a}^{60} - 1$

To Find:

• We have to show whether (a-1) is a factor of f(a) or not

Solution:

To show that (a-1) is a factor of f(a)

For an instance if (a-1) is a factor of f(a)

=> a - 1 = 0

=> a = 1

∴ a = 1 must be a zero of Polynomial f(a)

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$f(a) = {a}^{100} + {a}^{90} + {a}^{80} + {a}^{70} + {a}^{60} - 1$

Putting a = 1 in f(a)

$f(1) = {1}^{100} + {1}^{90} + {1}^{80} + {1}^{70} + {1}^{60} - 1$

$f(1) = 1 + 1 + 1 - 1 - 1 - 1$

$f(1) = 3 - 3$

$f(1) = 0$

Now it is clear that a = 1 is a zero of f(a)

Hence it is proved that (a-1) is a factor a factor of f(a)

_________________________

NOTE:

• One raise to the power any number is always equal to 1 .
• ${1}^{(any \: number)} = 1$