Combine (3z-7)/(z^2-4z-32)-(z+8)/(z^2-16)

Math
3z-7z2-4z-32-z+8z2-16
Simplify each term.
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Factor z2-4z-32 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 -32 and whose sum is -4.
-8,4
Write the factored form using these integers.
3z-7(z-8)(z+4)-z+8z2-16
3z-7(z-8)(z+4)-z+8z2-16
Simplify the denominator.
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Rewrite 16 as 42.
3z-7(z-8)(z+4)-z+8z2-42
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=z and b=4.
3z-7(z-8)(z+4)-z+8(z+4)(z-4)
3z-7(z-8)(z+4)-z+8(z+4)(z-4)
3z-7(z-8)(z+4)-z+8(z+4)(z-4)
To write 3z-7(z-8)(z+4) as a fraction with a common denominator, multiply by z-4z-4.
3z-7(z-8)(z+4)⋅z-4z-4-z+8(z+4)(z-4)
To write -z+8(z+4)(z-4) as a fraction with a common denominator, multiply by z-8z-8.
3z-7(z-8)(z+4)⋅z-4z-4-z+8(z+4)(z-4)⋅z-8z-8
Write each expression with a common denominator of (z-8)(z+4)(z-4), by multiplying each by an appropriate factor of 1.
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Multiply 3z-7(z-8)(z+4) and z-4z-4.
(3z-7)(z-4)(z-8)(z+4)(z-4)-z+8(z+4)(z-4)⋅z-8z-8
Multiply z+8(z+4)(z-4) and z-8z-8.
(3z-7)(z-4)(z-8)(z+4)(z-4)-(z+8)(z-8)(z+4)(z-4)(z-8)
Reorder the factors of (z-8)(z+4)(z-4).
(3z-7)(z-4)(z+4)(z-4)(z-8)-(z+8)(z-8)(z+4)(z-4)(z-8)
(3z-7)(z-4)(z+4)(z-4)(z-8)-(z+8)(z-8)(z+4)(z-4)(z-8)
Combine the numerators over the common denominator.
(3z-7)(z-4)-(z+8)(z-8)(z+4)(z-4)(z-8)
Simplify the numerator.
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Expand (3z-7)(z-4) using the FOIL Method.
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Apply the distributive property.
3z(z-4)-7(z-4)-(z+8)(z-8)(z+4)(z-4)(z-8)
Apply the distributive property.
3z⋅z+3z⋅-4-7(z-4)-(z+8)(z-8)(z+4)(z-4)(z-8)
Apply the distributive property.
3z⋅z+3z⋅-4-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
3z⋅z+3z⋅-4-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
Simplify and combine like terms.
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Simplify each term.
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Multiply z by z by adding the exponents.
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Move z.
3(z⋅z)+3z⋅-4-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
Multiply z by z.
3z2+3z⋅-4-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
3z2+3z⋅-4-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
Multiply -4 by 3.
3z2-12z-7z-7⋅-4-(z+8)(z-8)(z+4)(z-4)(z-8)
Multiply -7 by -4.
3z2-12z-7z+28-(z+8)(z-8)(z+4)(z-4)(z-8)
3z2-12z-7z+28-(z+8)(z-8)(z+4)(z-4)(z-8)
Subtract 7z from -12z.
3z2-19z+28-(z+8)(z-8)(z+4)(z-4)(z-8)
3z2-19z+28-(z+8)(z-8)(z+4)(z-4)(z-8)
Apply the distributive property.
3z2-19z+28+(-z-1⋅8)(z-8)(z+4)(z-4)(z-8)
Multiply -1 by 8.
3z2-19z+28+(-z-8)(z-8)(z+4)(z-4)(z-8)
Expand (-z-8)(z-8) using the FOIL Method.
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Apply the distributive property.
3z2-19z+28-z(z-8)-8(z-8)(z+4)(z-4)(z-8)
Apply the distributive property.
3z2-19z+28-z⋅z-z⋅-8-8(z-8)(z+4)(z-4)(z-8)
Apply the distributive property.
3z2-19z+28-z⋅z-z⋅-8-8z-8⋅-8(z+4)(z-4)(z-8)
3z2-19z+28-z⋅z-z⋅-8-8z-8⋅-8(z+4)(z-4)(z-8)
Simplify and combine like terms.
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Simplify each term.
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Multiply z by z by adding the exponents.
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Move z.
3z2-19z+28-(z⋅z)-z⋅-8-8z-8⋅-8(z+4)(z-4)(z-8)
Multiply z by z.
3z2-19z+28-z2-z⋅-8-8z-8⋅-8(z+4)(z-4)(z-8)
3z2-19z+28-z2-z⋅-8-8z-8⋅-8(z+4)(z-4)(z-8)
Multiply -8 by -1.
3z2-19z+28-z2+8z-8z-8⋅-8(z+4)(z-4)(z-8)
Multiply -8 by -8.
3z2-19z+28-z2+8z-8z+64(z+4)(z-4)(z-8)
3z2-19z+28-z2+8z-8z+64(z+4)(z-4)(z-8)
Subtract 8z from 8z.
3z2-19z+28-z2+0+64(z+4)(z-4)(z-8)
Add -z2 and 0.
3z2-19z+28-z2+64(z+4)(z-4)(z-8)
3z2-19z+28-z2+64(z+4)(z-4)(z-8)
Subtract z2 from 3z2.
2z2-19z+28+64(z+4)(z-4)(z-8)
Add 28 and 64.
2z2-19z+92(z+4)(z-4)(z-8)
2z2-19z+92(z+4)(z-4)(z-8)
Combine (3z-7)/(z^2-4z-32)-(z+8)/(z^2-16)

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