Question

the polynomials ax^3 - 3x^2 + 4 and 2x^3 - 5x + a when divided by (x-2) leave the remainder p and q respectively. if p - 4q = 4, find the value of 'a'

Answer 1

Answer:

It is given that a polynomial ax³ - 3x² + 4 when divided by (x - 2) leaves the remainder p. Let us substitute x = 2 in ax³ - 3x² + 4 and equate it to p as follows:

a(2)³ - 3(2)² +4=p

⇒ (a x 8) (3 x 4) + 4 = p

⇒ 8a12+4=p

⇒8a-8p....(1)

Similarly, given a polynomial 2x³ – 5x + a when divided by (x - 2) leaves the remainder q. Let us substitute x = 2 in 2x³ - 5x + a and equate it to q as follows:

2(2)³ (5 x 2) +a=q - ➡16 10+a=q ⇒6+a=q .... (2)

⇒ (2 x 8) - 10+ a=q

It is also given that p - 2q = 4, we, now substitute the values of p and q from equations 1 and 2 as shown below:

P-2q = 4

⇒8a-8-2(6+a)=4

8a-8-12-2a = 4

⇒8a-8-2(6+a)=4

⇒ 8a 8 12

→ 6a

20= 4

⇒ 6a 4+20

→6a = 24

24 6 ⇒a=

⇒a=4

Hence, a = 4.