Answer to Question #256002 in Algebra for iman munaim

Question

Given P(x) = x³ +ax² + bx + 26 has no remainder when divided by (x-1) and leaves remainder 32 when divided by (x-3).

(a)Find the values of a and b

(b)Hence, factorize P(x) completely

Answer 1

Given, (x³+ax²+bx+26) has (x−1) as a factor and leaves a remainder 32 when divided by (x−3).

(a)

Let f(x)=(x³+ax²+bx+26).

Since, (x−1) in a factor of f(x).

⇒f(1)=0

⇒1³+1²a+b+26=0

⇒1+a+b+26=0

⇒a+b+27=0 -----------------(1)

Also given, f(x) leaves a remainder 32 when divided by (x−3).

Therefore,

f(x−3)=32 (Remainder)

⇒f(3)=32

⇒3³+3²a+3b+26=32

⇒27+9a+3b+26=32

⇒3a+b+7=0 --------------(2)

By solving (1) & (2), we obtain,

b=−a−27 & b=−7−3a ----------(3)

Equating the value of b,

−a−27=−7−3a

3a−a=−7+27

a=10

And, from (3)

b=−(a)−27=-(10)−27=-37

b=−37

(b)

factorise f(x)=(x³+10x²-37x+26),

x³-x²+11x²-11x-26x+26=0

x²(x-1)+11x(x-1)-26(x-1)=0

(x-1)(x²+11x-26)=0

(x-1)(x²-2x+13x-26)=0

(x-1)(x(x-2)+13(x-2))=0

(x-1)(x-2)(x+13)=0