Greatest Common Divisor of Two Polynomials

You find the greatest common divisor of two polynomials in the same way as you found the greatest common divisorDM2-2.tex#Greatest common divisor of two integers.

$\blacktriangleright$ To find the greatest common divisor of two or more polynomials

1.
Type gcd in mathematics mode, or choose it from the functions menu.

2.
Click itbpF0.3009in0.3009in0.0701inparens.wmf and type the polynomials in mathematics mode, separated by red commas.

3.
Choose Evaluate.

$\blacktriangleright$ Evaluate

gcd(5x2 - 5x, 10x - 10) = 5x - 5

gcd$\left(\vphantom{ yx+3x-5y-15,\,xz-53x-5z+265}\right.$yx + 3x - 5y - 15, xz - 53x - 5z + 265$\left.\vphantom{ yx+3x-5y-15,\,xz-53x-5z+265}\right)$ =  x - 5

gcd$\left(\vphantom{ x^{2}+3x+yx+3y,\,x^{2}-4yx-5y^{2},3x^{2}+2yx-y^{2}}\right.$x2 +3x + yx + 3y, x2 -4yx - 5y2, 3x2 +2yx - y2$\left.\vphantom{ x^{2}+3x+yx+3y,\,x^{2}-4yx-5y^{2},3x^{2}+2yx-y^{2}}\right)$ = x + y

You can check these results by factoring the polynomials and comparing the factors.

$\blacktriangleright$ Factor

5x2 -5x = 5x$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$

10x - 10 = 10$\left(\vphantom{ x-1}\right.$x - 1$\left.\vphantom{ x-1}\right)$

yx + 3x - 5y - 15 = $\left(\vphantom{ x-5}\right.$x - 5$\left.\vphantom{ x-5}\right)$$\left(\vphantom{ y+3}\right.$y + 3$\left.\vphantom{ y+3}\right)$

xz - 53x - 5z + 265 = $\left(\vphantom{ x-5}\right.$x - 5$\left.\vphantom{ x-5}\right)$$\left(\vphantom{ z-53}\right.$z - 53$\left.\vphantom{ z-53}\right)$

x2 +3x + yx + 3y = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ x+3}\right.$x + 3$\left.\vphantom{ x+3}\right)$

x2 -4yx - 5y2 = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ x-5y}\right.$x - 5y$\left.\vphantom{ x-5y}\right)$

3x2 +2yx - y2 = $\left(\vphantom{ x+y}\right.$x + y$\left.\vphantom{ x+y}\right)$$\left(\vphantom{ 3x-y}\right.$3x - y$\left.\vphantom{ 3x-y}\right)$

The least common multipleDM2-2.tex#Greatest common divisor function is also available for polynomials.

$\blacktriangleright$ To find the least common multiple of two or more polynomials

1.
Type lcm in mathematics mode. (It will turn gray.)

1.
Click itbpF0.3009in0.3009in0.0701inparens.wmf or choose Insert + Brackets and select parentheses.

2.
In the input box, type the polynomials separated by (red) commas.

3.
Choose Evaluate.

Apply Factor to the result to reveal the relationship among the polynomials.

$\blacktriangleright$ Evaluate, Factor

$\limfunc$lcm(yx + 3x - 5y - 15, xz - 53x - 5z + 265)

     = xyz - 53xy + 3xz - 159x - 5yz + 265y - 15z + 795

     = $\left(\vphantom{ x-5}\right.$x - 5$\left.\vphantom{ x-5}\right)$$\left(\vphantom{ y+3}\right.$y + 3$\left.\vphantom{ y+3}\right)$$\left(\vphantom{ z-53}\right.$z - 53$\left.\vphantom{ z-53}\right)$