Simplify ((4y^2-9)/(2y^2+11y-21))÷((2y^2+y-3)/(y^2+6y-7))

Math
4y2-92y2+11y-21÷2y2+y-3y2+6y-7
To divide by a fraction, multiply by its reciprocal.
4y2-92y2+11y-21⋅y2+6y-72y2+y-3
Simplify the numerator.
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Rewrite 4y2 as (2y)2.
(2y)2-92y2+11y-21⋅y2+6y-72y2+y-3
Rewrite 9 as 32.
(2y)2-322y2+11y-21⋅y2+6y-72y2+y-3
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2y and b=3.
(2y+3)(2y-3)2y2+11y-21⋅y2+6y-72y2+y-3
(2y+3)(2y-3)2y2+11y-21⋅y2+6y-72y2+y-3
Factor by grouping.
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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⋅-21=-42 and whose sum is b=11.
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Factor 11 out of 11y.
(2y+3)(2y-3)2y2+11(y)-21⋅y2+6y-72y2+y-3
Rewrite 11 as -3 plus 14
(2y+3)(2y-3)2y2+(-3+14)y-21⋅y2+6y-72y2+y-3
Apply the distributive property.
(2y+3)(2y-3)2y2-3y+14y-21⋅y2+6y-72y2+y-3
(2y+3)(2y-3)2y2-3y+14y-21⋅y2+6y-72y2+y-3
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
(2y+3)(2y-3)(2y2-3y)+14y-21⋅y2+6y-72y2+y-3
Factor out the greatest common factor (GCF) from each group.
(2y+3)(2y-3)y(2y-3)+7(2y-3)⋅y2+6y-72y2+y-3
(2y+3)(2y-3)y(2y-3)+7(2y-3)⋅y2+6y-72y2+y-3
Factor the polynomial by factoring out the greatest common factor, 2y-3.
(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3
(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3
Cancel the common factor of 2y-3.
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Cancel the common factor.
(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3
Rewrite the expression.
2y+3y+7⋅y2+6y-72y2+y-3
2y+3y+7⋅y2+6y-72y2+y-3
Factor y2+6y-7 using the AC method.
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Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is -7 and whose sum is 6.
-1,7
Write the factored form using these integers.
2y+3y+7⋅(y-1)(y+7)2y2+y-3
2y+3y+7⋅(y-1)(y+7)2y2+y-3
Factor by grouping.
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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⋅-3=-6 and whose sum is b=1.
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Multiply by 1.
2y+3y+7⋅(y-1)(y+7)2y2+1y-3
Rewrite 1 as -2 plus 3
2y+3y+7⋅(y-1)(y+7)2y2+(-2+3)y-3
Apply the distributive property.
2y+3y+7⋅(y-1)(y+7)2y2-2y+3y-3
2y+3y+7⋅(y-1)(y+7)2y2-2y+3y-3
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
2y+3y+7⋅(y-1)(y+7)(2y2-2y)+3y-3
Factor out the greatest common factor (GCF) from each group.
2y+3y+7⋅(y-1)(y+7)2y(y-1)+3(y-1)
2y+3y+7⋅(y-1)(y+7)2y(y-1)+3(y-1)
Factor the polynomial by factoring out the greatest common factor, y-1.
2y+3y+7⋅(y-1)(y+7)(y-1)(2y+3)
2y+3y+7⋅(y-1)(y+7)(y-1)(2y+3)
Cancel the common factor of 2y+3.
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Factor 2y+3 out of (y-1)(2y+3).
2y+3y+7⋅(y-1)(y+7)(2y+3)(y-1)
Cancel the common factor.
2y+3y+7⋅(y-1)(y+7)(2y+3)(y-1)
Rewrite the expression.
1y+7⋅(y-1)(y+7)y-1
1y+7⋅(y-1)(y+7)y-1
Cancel the common factor of y+7.
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Factor y+7 out of (y-1)(y+7).
1y+7⋅(y+7)(y-1)y-1
Cancel the common factor.
1y+7⋅(y+7)(y-1)y-1
Rewrite the expression.
y-1y-1
y-1y-1
Cancel the common factor of y-1.
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Cancel the common factor.
y-1y-1
Divide 1 by 1.
1
1
Simplify ((4y^2-9)/(2y^2+11y-21))÷((2y^2+y-3)/(y^2+6y-7))

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