# Simplify ((3y^2*(13y)+4)/(y^2-16))÷((4y^2-1)/(2y^2-9y+4))

3y2⋅(13y)+4y2-16÷4y2-12y2-9y+4
To divide by a fraction, multiply by its reciprocal.
3y2⋅(13y)+4y2-16⋅2y2-9y+44y2-1
Simplify the numerator.
Rewrite using the commutative property of multiplication.
3⋅13(y2y)+4y2-16⋅2y2-9y+44y2-1
Multiply y2 by y by adding the exponents.
Multiply y2 by y.
Raise y to the power of 1.
3⋅13(y2y1)+4y2-16⋅2y2-9y+44y2-1
Use the power rule aman=am+n to combine exponents.
3⋅13y2+1+4y2-16⋅2y2-9y+44y2-1
3⋅13y2+1+4y2-16⋅2y2-9y+44y2-1
3⋅13y3+4y2-16⋅2y2-9y+44y2-1
3⋅13y3+4y2-16⋅2y2-9y+44y2-1
Multiply 3 by 13.
39y3+4y2-16⋅2y2-9y+44y2-1
39y3+4y2-16⋅2y2-9y+44y2-1
Simplify the denominator.
Rewrite 16 as 42.
39y3+4y2-42⋅2y2-9y+44y2-1
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=4.
39y3+4(y+4)(y-4)⋅2y2-9y+44y2-1
39y3+4(y+4)(y-4)⋅2y2-9y+44y2-1
Factor by grouping.
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=2⋅4=8 and whose sum is b=-9.
Factor -9 out of -9y.
39y3+4(y+4)(y-4)⋅2y2-9(y)+44y2-1
Rewrite -9 as -1 plus -8
39y3+4(y+4)(y-4)⋅2y2+(-1-8)y+44y2-1
Apply the distributive property.
39y3+4(y+4)(y-4)⋅2y2-1y-8y+44y2-1
39y3+4(y+4)(y-4)⋅2y2-1y-8y+44y2-1
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
39y3+4(y+4)(y-4)⋅(2y2-1y)-8y+44y2-1
Factor out the greatest common factor (GCF) from each group.
39y3+4(y+4)(y-4)⋅y(2y-1)-4(2y-1)4y2-1
39y3+4(y+4)(y-4)⋅y(2y-1)-4(2y-1)4y2-1
Factor the polynomial by factoring out the greatest common factor, 2y-1.
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)4y2-1
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)4y2-1
Simplify the denominator.
Rewrite 4y2 as (2y)2.
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)(2y)2-1
Rewrite 1 as 12.
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)(2y)2-12
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2y and b=1.
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)(2y+1)(2y-1)
39y3+4(y+4)(y-4)⋅(2y-1)(y-4)(2y+1)(2y-1)
Simplify terms.
Cancel the common factor of y-4.
Factor y-4 out of (y+4)(y-4).
39y3+4(y-4)(y+4)⋅(2y-1)(y-4)(2y+1)(2y-1)
Factor y-4 out of (2y-1)(y-4).
39y3+4(y-4)(y+4)⋅(y-4)(2y-1)(2y+1)(2y-1)
Cancel the common factor.
39y3+4(y-4)(y+4)⋅(y-4)(2y-1)(2y+1)(2y-1)
Rewrite the expression.
39y3+4y+4⋅2y-1(2y+1)(2y-1)
39y3+4y+4⋅2y-1(2y+1)(2y-1)
Multiply 39y3+4y+4 and 2y-1(2y+1)(2y-1).
(39y3+4)(2y-1)(y+4)((2y+1)(2y-1))
Cancel the common factor of 2y-1.
Cancel the common factor.
(39y3+4)(2y-1)(y+4)((2y+1)(2y-1))
Rewrite the expression.
39y3+4(y+4)(2y+1)
39y3+4(y+4)(2y+1)
39y3+4(y+4)(2y+1)
Simplify ((3y^2*(13y)+4)/(y^2-16))÷((4y^2-1)/(2y^2-9y+4))