Greatest Common Divisor of Polynomials

The greatest common divisor of two polynomials p(x) and q(x) is a polynomial d (x) of highest degree that divides both p(x) and q(x).

Define p(x) = 18x7 -9x5 +36x4 +4x3 -16x2 + 19x + 12 and q(x) = 15x5 -9x4 +11x3 +17x2 - 10x + 8, then use Evaluate to calculate gcd$\left(\vphantom{ p(x),q(x)}\right.$p(x), q(x)$\left.\vphantom{ p(x),q(x)}\right)$.


$\blacktriangleright$ Evaluate

gcd$\left(\vphantom{ p(x),q(x)}\right.$p(x), q(x)$\left.\vphantom{ p(x),q(x)}\right)$ = 3x3 + x + 4


Use the following to verify that 3x3 + x + 4 is indeed a common divisor.


$\blacktriangleright$ Polynomials + Divide

${\dfrac{{18x^{7}-9x^{5}+36x^{4}-5x^{3}-16x^{2}+16x}}{{3x^{3}+x+4}%
}}$ = 6x4 -5x2 +4x


${\dfrac{{15x^{5}-9x^{4}+11x^{3}+17x^{2}-10x+8}}{{3x^{3}+x+4}}}$ = 5x2 -3x + 2


Thus,

p(x) = $\displaystyle \left(\vphantom{ 6x^4-5x^2+4x+3}\right.$6x4 -5x2 + 4x + 3$\displaystyle \left.\vphantom{ 6x^4-5x^2+4x+3}\right)$$\displaystyle \left(\vphantom{ 3x^3+x+4}\right.$3x3 + x + 4$\displaystyle \left.\vphantom{ 3x^3+x+4}\right)$

and

q(x) = $\displaystyle \left(\vphantom{ 5x^2-3x+2}\right.$5x2 - 3x + 2$\displaystyle \left.\vphantom{ 5x^2-3x+2}\right)$$\displaystyle \left(\vphantom{ 3x^3+x+4}\right.$3x3 + x + 4$\displaystyle \left.\vphantom{ 3x^3+x+4}\right)$