Help me to solve this tough math problem!!?

Question

If P(x) is a polynomial in x, and x^23 + 23x^17 – 18x^16 – 24x^15 + 108x^14 = (x^4 – 3x^2 – 2x + 9). P(x) for all values of x. Compute the sum of the coefficient x.

the answer is : 18.

kindly explain it to me how to obtain 18..

thanks

Answer 1

do u seriouly expect someone to take their time to figure this out for u?

Answer 2

I am assuming your equation is...

x^23 + 23x^17 – 18x^16 – 24x^15 + 108x^14

=(x^4 – 3x^2 – 2x + 9)*P(x)

...and that the question is asking you to find the sum of the coefficients of x in P(x).

Firstly, to obtain P(x), divide both sides by (x^4 – 3x^2 – 2x + 9). Factorising...

P(x)=(x^23 + 23x^17 – 18x^16 – 24x^15 + 108x^14) /(x^4 – 3x^2 – 2x + 9)

=[x^14 (x^9 + x^23 - 18x^2 - 24^x + 108)] / (x^4 - 3x^2 - 2x + 9)

=x^14 (x^5 + 3x^3 + 2x^2 + 12)

Adding the coefficients, 1+3+2+12=18

(I'm not sure whether long division of polynomials is the most elegant solution... however, it's the only one I can think of currently)

See http://www.sosmath.com/algebra/factor/fa... the third subsection on how to perform the polynomial long division (less painful than writing out the entire division long hand, avoiding division with x^8, x^7 etc... all having coefficients of zero)