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Answer 1

Answer:

a = 1, when R1 = R2.  a = -2.354 when R1 + R2 = 0. a = 0.1417, when 2R1 - R2 = 0.

Step-by-step explanation:

f(x) = a$x^{3}$ + 3$x^{2}$ - 3

f(x) when divided by (x - 4) leaves remainder $R_{1}$

i.e. when f(x) - $R_{1}$ is divided by (x -4), remainder will be zero.

⇒ (x-4) = 0, is a factor of f(x) - $R_{1}$

⇒ x = 4, is a factor of f(x) - $R_{1}$

f(x) - $R_{1}$ = 0, for x = 4

a $x^{3}$ + 3$x^{2}$ - 3 - $R_{1}$ = 0, for x = 4

⇒ a × $4^{3}$ + 3 × $4^{2}$ - 3 - $R_{1}$ = 0

64a + 3 × 16 - 3 - $R_{1}$ = 0

$R_{1}$ = 64a + 48 - 3 = 64a + 45

similarly,

g(x) - $R_{2}$ = 0, for x = 4

2$x^{3}$  - 5$x$ + a - $R_{2}$ = 0, for x = 4

2 × $4^{3}$  - 5 × 4 + a - $R_{2}$ = 0 = 0

2 × 64  - 20 + a - $R_{2}$ = 0

$R_{2}$ = 128 - 20 + a = 108 + a

when $R_{1}$ = $R_{2}$,

64a + 45 = 108 + a

64a - a = 108 - 45

63a = 63, ⇒ a = 1

when $R_{1}$ + $R_{2}$ = 0,

(i.e. when $R_{1}$ = - $R_{2}$)

64a + 45 = -(108 + a)

64a + 45 = -108 - a

64a + a = -108 -45

65a = -(108 + 45)

65a = -153

a = $\frac{-153}{65}$

a = - ( $2\frac{23}{65}$ )

a = - ( $2 + \frac{23}{65}$ )

a = - 2.354

(153 when divided by 65 leaves a remainder 23, quotient = 2)

(answer = quotient + (remainder/divisor), 153 = 2 + (23/65)

when $2R_{1}$ - $R_{2}$ = 0,

(i.e. $R_{2}$ = $2R_{1}$)

108 + a = 2 × (64a + 45)

108 + a = 2 × 64a + 2 × 45

108 + a = 128a + 90

128a - a = 108 - 90

127a = 18

a = $\frac{18}{127}$ ≅ 0.1417