Simplify ((12n^2-363)/(2n^2-25n+77))÷((14n^2+73n-22)/(n^2-15n+56))

Math
12n2-3632n2-25n+77÷14n2+73n-22n2-15n+56
To divide by a fraction, multiply by its reciprocal.
12n2-3632n2-25n+77⋅n2-15n+5614n2+73n-22
Simplify the numerator.
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Factor 3 out of 12n2-363.
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Factor 3 out of 12n2.
3(4n2)-3632n2-25n+77⋅n2-15n+5614n2+73n-22
Factor 3 out of -363.
3(4n2)+3(-121)2n2-25n+77⋅n2-15n+5614n2+73n-22
Factor 3 out of 3(4n2)+3(-121).
3(4n2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22
3(4n2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22
Rewrite 4n2 as (2n)2.
3((2n)2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22
Rewrite 121 as 112.
3((2n)2-112)2n2-25n+77⋅n2-15n+5614n2+73n-22
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2n and b=11.
3(2n+11)(2n-11)2n2-25n+77⋅n2-15n+5614n2+73n-22
3(2n+11)(2n-11)2n2-25n+77⋅n2-15n+5614n2+73n-22
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⋅77=154 and whose sum is b=-25.
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Factor -25 out of -25n.
3(2n+11)(2n-11)2n2-25(n)+77⋅n2-15n+5614n2+73n-22
Rewrite -25 as -11 plus -14
3(2n+11)(2n-11)2n2+(-11-14)n+77⋅n2-15n+5614n2+73n-22
Apply the distributive property.
3(2n+11)(2n-11)2n2-11n-14n+77⋅n2-15n+5614n2+73n-22
3(2n+11)(2n-11)2n2-11n-14n+77⋅n2-15n+5614n2+73n-22
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
3(2n+11)(2n-11)(2n2-11n)-14n+77⋅n2-15n+5614n2+73n-22
Factor out the greatest common factor (GCF) from each group.
3(2n+11)(2n-11)n(2n-11)-7(2n-11)⋅n2-15n+5614n2+73n-22
3(2n+11)(2n-11)n(2n-11)-7(2n-11)⋅n2-15n+5614n2+73n-22
Factor the polynomial by factoring out the greatest common factor, 2n-11.
3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22
3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22
Cancel the common factor of 2n-11.
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Cancel the common factor.
3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22
Rewrite the expression.
3(2n+11)n-7⋅n2-15n+5614n2+73n-22
3(2n+11)n-7⋅n2-15n+5614n2+73n-22
Factor n2-15n+56 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 56 and whose sum is -15.
-8,-7
Write the factored form using these integers.
3(2n+11)n-7⋅(n-8)(n-7)14n2+73n-22
3(2n+11)n-7⋅(n-8)(n-7)14n2+73n-22
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=14⋅-22=-308 and whose sum is b=73.
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Factor 73 out of 73n.
3(2n+11)n-7⋅(n-8)(n-7)14n2+73(n)-22
Rewrite 73 as -4 plus 77
3(2n+11)n-7⋅(n-8)(n-7)14n2+(-4+77)n-22
Apply the distributive property.
3(2n+11)n-7⋅(n-8)(n-7)14n2-4n+77n-22
3(2n+11)n-7⋅(n-8)(n-7)14n2-4n+77n-22
Factor out the greatest common factor from each group.
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Group the first two terms and the last two terms.
3(2n+11)n-7⋅(n-8)(n-7)(14n2-4n)+77n-22
Factor out the greatest common factor (GCF) from each group.
3(2n+11)n-7⋅(n-8)(n-7)2n(7n-2)+11(7n-2)
3(2n+11)n-7⋅(n-8)(n-7)2n(7n-2)+11(7n-2)
Factor the polynomial by factoring out the greatest common factor, 7n-2.
3(2n+11)n-7⋅(n-8)(n-7)(7n-2)(2n+11)
3(2n+11)n-7⋅(n-8)(n-7)(7n-2)(2n+11)
Cancel the common factor of 2n+11.
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Factor 2n+11 out of 3(2n+11).
(2n+11)⋅3n-7⋅(n-8)(n-7)(7n-2)(2n+11)
Factor 2n+11 out of (7n-2)(2n+11).
(2n+11)⋅3n-7⋅(n-8)(n-7)(2n+11)(7n-2)
Cancel the common factor.
(2n+11)⋅3n-7⋅(n-8)(n-7)(2n+11)(7n-2)
Rewrite the expression.
3n-7⋅(n-8)(n-7)7n-2
3n-7⋅(n-8)(n-7)7n-2
Cancel the common factor of n-7.
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Factor n-7 out of (n-8)(n-7).
3n-7⋅(n-7)(n-8)7n-2
Cancel the common factor.
3n-7⋅(n-7)(n-8)7n-2
Rewrite the expression.
3n-87n-2
3n-87n-2
Combine 3 and n-87n-2.
3(n-8)7n-2
Simplify ((12n^2-363)/(2n^2-25n+77))÷((14n^2+73n-22)/(n^2-15n+56))

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