Question

If p(x) is a polynomial of degree 3 with leading coefficient 1006. Also p(1)=1, p(2)=3, p(3)=5. What is the value of p(5) :

Answer 1

Given: p(x) is a polynomial of degree 3 with leading coefficient 1006.

To find: The value of p(5)

Solution:

• Now we have given p(x) of degree 3, it has leading coefficient 1006.

• We have provided with p (1) = 1, p (2) = 3, p(3) = 5

• Let the equation be:

                  p(x) = 1006x³  + bx² + cx  + d

• Now p(1) will be:

                  p(1) =  1006 + b + c + d = 1

                  b + c + d = -1005   ....................(i)

• p(2) will be:

                  p(2) = 1006(8) + 4b + 2c + d = 3

                  4b + 2c + d = -8045 ....................(ii)

• p(3) will be:

                  p(3) =  1006(27) + 9b + 3c + d = 5

                  9b + 3c + d = -27157 ....................(iii)

• Now, (ii) - (i), we get:

                  3b + c  = -7040....................(iv)

• Now, (iii) - (ii), we get:

                  5b + c  =  -19112....................(v)

• Now, (v) - (iv), we get:

                  2b  =  -12072

                  b  =  -6036

• Putting b in (iv), we get:

                  c = 11068

• Putting b and c in (i), we get:

                  d  =  -6037

• So the equation formed is:

                  p(x) = 1006x³ - 6036 x² + 11068x - 6037

• Now p(5) will be:

                  p(5) = 1006(5)³ - 6036(5)² + 11068(5) - 6037

                  p(5) = 24153

Answer:

           So the value of p(5) is 24153.