In each of the following two polynomials, find the value of a, if x-a

1.	In each of the following two polynomials, find the value of a, if x-a is a factor:(i) x6 - ax5 + x4 - ax3 + 3x - a + 2.(ii) x5 - a2x3 + 2x + a + 1.
Answer» (i) Let f (x) = x⁶ - ax⁵ + x⁴ - ax³ + 3x-a + 2 be the given polynomial From factor theorem, If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a] f (a) = 0 (a)⁶– a (a)⁵ + (a)⁴ – a (a)³ + 3 (a) – a + 2 = 0 a⁶ – a⁶ + a⁴ – a⁴ + 3a – a + 2 = 0 2a + 2 = 0 a = -1 Hence, (x – a) is a factor f (x) when a = -1. (ii) Let, f (x) = x⁵ - a²x³ + 2x + a + 1 be the given polynomial From factor theorem, If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a] f (a) = 0 (a)⁵– a² (a)³ + 2 (a) + a + 1 = 0 a⁵ – a⁵ + 2a + a + 1 = 0 3a + 1 = 0 3a = -1 a = \(\frac{-1}{3}\) Hence, (x – a) is a factor f (x) when a = \(\frac{-1}{3}.\)

In each of the following two polynomials, find the value of a, if x-a is a factor:(i) x6 - ax5 + x4 - ax3 + 3x - a + 2.(ii) x5 - a2x3 + 2x + a + 1.

Answer»

(i) Let f (x) = x⁶ - ax⁵ + x⁴ - ax³ + 3x-a + 2 be the given polynomial

From factor theorem,

If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a]

f (a) = 0

(a)⁶– a (a)⁵ + (a)⁴ – a (a)³ + 3 (a) – a + 2 = 0

a⁶ – a⁶ + a⁴ – a⁴ + 3a – a + 2 = 0

2a + 2 = 0

a = -1

Hence,

(x – a) is a factor f (x) when a = -1.

(ii) Let, f (x) = x⁵ - a²x³ + 2x + a + 1 be the given polynomial

From factor theorem,

If (x – a) is a factor of f (x) then f (a) = 0 [Therefore, x – a = 0, x = a]

f (a) = 0

(a)⁵– a² (a)³ + 2 (a) + a + 1 = 0

a⁵ – a⁵ + 2a + a + 1 = 0

3a + 1 = 0

3a = -1

a = \(\frac{-1}{3}\)

Hence,

(x – a) is a factor f (x) when a = \(\frac{-1}{3}.\)