Polynomials (again) (1 Viewer)

[taggs-sasuke]

Hey, this time I can't get the answer provided in the solutions.

The question is:

Calculate the values of p, q, r, s in order that

2x^3 +4x^2 + 5x + 6

and

p(x-1)^3 + q(x-1)^2 + r(x-1) + s

specify the same polynomial.

The answer is:

p=2, q=10, r=19, s=17

I have different answers to the ones in bold.

Are the answers wrong or did I make a really stupid mistake (sigh)?

Thank you!

 

[annabackwards]

Just equate the coefficients of x^3, x^2, x and equate the constants.
For the answers that you got wrong in bold, make sure you don't forget the other x^2 that come from expanding out p(x-1)^3 and q(x-1)^2 - that's probably why you got the wrong answers.

Since
p(x - 1)^3
= p(x - 1)(x - 1)(x - 1) = p(x - 1)(x^2 - 2x + 1)
= px^3 - p3x^2 + p3x - p

q(x-1)^2
= q(x^2 - 2x +1)
= qx^2 - q2x + q

Equating the coefficients of x^2 gives
q - 3p = 4
q - 6 = 4
= 10

Equating the coefficients of x gives
r + 3p - 2q = 5
r = 5 - 3p + 2q
= 5 - 6 + 20
= 19

 

[taggs-sasuke]

:mad1: Thank you!

 

[annabackwards]

Your welcome