Q8 of 82 Page 130

Let us show that if n is any positive integer (even or odd), x – y is a factor of the polynomial xn – yn.

Formula used.


If f(x) is a polynomial with degree n


Then (x – a) is a factor of f(x) if f(a) = 0


⇒ Dividend = Divisor × Quotient + Remainder


If (x – y) is a factor of xn – yn


Then


We have to prove


On dividing xn – yn by (x – y) Remainder gets 0


When n = 1


(x – y) becomes factor of (x1 – y1)


Hence,


x – y = 0


x = y


Suppose on dividing xn – yn with x – y we get Remainder R


xn – yn = (x – y) × Quotient + R


Putting x = y


yn – yn = (y – y) × Quotient + R


0 = 0 + R


R = 0


∴ For every value n (x – y) is a factor of xn – yn


Conclusion.


⇒ For every value n (x – y) is a factor of xn – yn


More from this chapter

All 82 →
6

If (x + 1) and (x + 2) are two factors of the polynomial x3 + 3x2 + 2ax + b, then let us calculate and write the values of a and b.

 7

If the polynomial ax3 + bx2 + x – 6 is divided by x – 2 and remainder is 4, then let us calculate the values of a and b when x + 2 is a factor of this polynomial.

 9

Let us show that if n is any positive odd integer, then x + y is a factor of xn + yn.

 10

Let us show that if n be any positive integer (even or odd), the x – y never be a factor of the polynomial xn + yn.