Question

The cubic polynomial p(x) satisfies the condition that (x-1)^2 is a factor of p(x)+2 and (x+1)^2 is a factor of p(x)-2. then p(3) equals

Answer 1

Given:

A cubic polynomial, P(x) = x³ + bx² + cx + d.

(x-1)^2 is a factor of p(x)+2 and

(x+1)^2 is a factor of p(x)-2

To Find:

P(3).

Solution:

Since x - 1 is a factor of P(x) + 2 , x  = 1 is a solution of P(x) + 2.

Also x + 1 is a factor  of P(x) - 2, x = -1 is a solution of P(x) - 2.

• P(x) +2 = (x−1²)(ax+b)  

• P(x) −2=(x+1²)(cx+d )

Lets  equate P(x)

• (x − 1)²(ax + b) −2 = (x + 1)² (cx + d) + 2

Expand ( x -1 )² and (x+1)²

• (x²−2x+1)(ax+b) −2=(x²+2x+1)(cx+d)+2

Multiplying all the terms,

• ax³+(b−2a)x²+(a−2b)x+b} −2 = cx³+(2c+d)x²+(2d+c)x+d}+2

Now we can compare the coefficients of different powers of x

    1. x³

• a = c  -----(1)

    2. x²

• b−2a = d+2c-----(2)

     3. x

• (a−2b)=(c+2d)-------(3)

     4. constant terms

• b −2=d+ 2 -----(4)

Form (4)  

•  b=d+4 ----(5)

From (2),

• b= d+2c+2a =  d+2a+2a = d+4a

Hence, d+4=d+4a    

• a =1 and  c= 1

From(3)

• a = c+2d+2b

Therefore,

• 1+2d+2b=1
•   b = −d

From (5)  

• b = d+4
• b= - b + 4
• b=2

• Hence  d =  -2

Therefore, the polynomial is  

• P(x) +2=(x−1)²(x+2)  
• P(x) + 2 = (x + 2)(x² -2x + 1)
• P(x) = x³-3x + 2 - 2

Therefore, p(x) =x³−3x

Then, p(3) =3³−3×3= 18

Answer 2

Answer:

18

Step-by-step explanation:

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