How do you find the greatest common factor of 45y^{12}+30y^{10}?

3 Answers
Jun 11, 2018

15y^10

Explanation:

The largest number that goes into 45 and 30 is 15

y^10 is the biggest term that goes into y^12 and y^10

Jun 11, 2018

The greatest common factor is 15y^10.

Explanation:

I suppose you're asking for the greatest common factor between 45y^12 and 30 y^10, since the greatest common factor is computed between two number.

We need to find the greatest common factor for both the numeric and literal part.

For the numeric part, we can use the prime factorization of the two numbers:

45 = 9*5 = 3^2*5

30 = 3*10 = 2*3*5

So, what's the biggest number that "fits" inside both 45 and 30, given their prime factorizations?

Well, we can't choose 2, because it fits in 30 but not in 45. 3 appears in both factorization, but we can pick it only once, since two three's (i.e. 9) fit inside 45, but not inside 30. Finally, we can pick 5 once since it appears in both factorizations.

So, the answer is 3*5 = 15

As for the literal part, we have a similar way to proceed: since "there are" 12 y's in the first term and only 10 in the second, we can take at most 10 y's from both.

So, the greatest common factor is 15y^10. In fact, you can factor it from both terms to get

45y^12 + 30 y^10 = 15y^10(3y^2+2)

and there is nothing else to factor between 3y^2 and 2.

Jun 11, 2018

15y^10(3y^2+2)

Broken down into steps

Explanation:

color(brown)("If you are not sure break it down into stages.")

We know that 5 is a factor of both 45 and 30

5(9y^12+6y^10)

We know that 3 is a factor of both 9 and 6

5[3(3y^12+2y^10)]

15[3y^12+2y^10]

We know that y^12->color(red)(y^10)color(green)(xxy^2) giving:

15[color(white)(2/2)color(white)("dddd")3y^12color(white)("ddd.d")+2y^10color(white)(2/2)]

color(green)(15[color(white)(2/2)obrace( (3color(red)(xxy^10)xxy^2))+2color(red)(y^10)color(white)(2/2)])

Factor out the color(red)(y^10) giving:

15y^10(3y^2+2)