# Combine t^2+3/(t^4-16)+7/(16-t^4)

t2+3t4-16+716-t4
Simplify each term.
Simplify the denominator.
Rewrite t4 as (t2)2.
t2+3(t2)2-16+716-t4
Rewrite 16 as 42.
t2+3(t2)2-42+716-t4
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=t2 and b=4.
t2+3(t2+4)(t2-4)+716-t4
Simplify.
Rewrite 4 as 22.
t2+3(t2+4)(t2-22)+716-t4
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=t and b=2.
t2+3(t2+4)(t+2)(t-2)+716-t4
t2+3(t2+4)(t+2)(t-2)+716-t4
t2+3(t2+4)(t+2)(t-2)+716-t4
Simplify the denominator.
Rewrite 16 as 42.
t2+3(t2+4)(t+2)(t-2)+742-t4
Rewrite t4 as (t2)2.
t2+3(t2+4)(t+2)(t-2)+742-(t2)2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=4 and b=t2.
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(4-t2)
Simplify.
Rewrite 4 as 22.
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(22-t2)
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2 and b=t.
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
To write t2 as a fraction with a common denominator, multiply by (t2+4)(t+2)(t-2)(t2+4)(t+2)(t-2).
t2⋅(t2+4)(t+2)(t-2)(t2+4)(t+2)(t-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Simplify terms.
Combine t2 and (t2+4)(t+2)(t-2)(t2+4)(t+2)(t-2).
t2((t2+4)(t+2)(t-2))(t2+4)(t+2)(t-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Combine the numerators over the common denominator.
t2((t2+4)(t+2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t2+4)(t+2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Simplify the numerator.
Expand (t2+4)(t+2) using the FOIL Method.
Apply the distributive property.
t2((t2(t+2)+4(t+2))(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Apply the distributive property.
t2((t2t+t2⋅2+4(t+2))(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Apply the distributive property.
t2((t2t+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t2t+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Simplify each term.
Multiply t2 by t by adding the exponents.
Multiply t2 by t.
Raise t to the power of 1.
t2((t2t1+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Use the power rule aman=am+n to combine exponents.
t2((t2+1+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t2+1+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t3+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t3+t2⋅2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Move 2 to the left of t2.
t2((t3+2⋅t2+4t+4⋅2)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply 4 by 2.
t2((t3+2t2+4t+8)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2((t3+2t2+4t+8)(t-2))+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Expand (t3+2t2+4t+8)(t-2) by multiplying each term in the first expression by each term in the second expression.
t2(t3t+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Simplify each term.
Multiply t3 by t by adding the exponents.
Multiply t3 by t.
Raise t to the power of 1.
t2(t3t1+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Use the power rule aman=am+n to combine exponents.
t2(t3+1+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t3+1+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4+t3⋅-2+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Move -2 to the left of t3.
t2(t4-2⋅t3+2t2t+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply t2 by t by adding the exponents.
Move t.
t2(t4-2t3+2(t⋅t2)+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply t by t2.
Raise t to the power of 1.
t2(t4-2t3+2(t1t2)+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Use the power rule aman=am+n to combine exponents.
t2(t4-2t3+2t1+2+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-2t3+2t1+2+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-2t3+2t3+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-2t3+2t3+2t2⋅-2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply -2 by 2.
t2(t4-2t3+2t3-4t2+4t⋅t+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply t by t by adding the exponents.
Move t.
t2(t4-2t3+2t3-4t2+4(t⋅t)+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply t by t.
t2(t4-2t3+2t3-4t2+4t2+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-2t3+2t3-4t2+4t2+4t⋅-2+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply -2 by 4.
t2(t4-2t3+2t3-4t2+4t2-8t+8t+8⋅-2)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply 8 by -2.
t2(t4-2t3+2t3-4t2+4t2-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-2t3+2t3-4t2+4t2-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Combine the opposite terms in t4-2t3+2t3-4t2+4t2-8t+8t-16.
t2(t4+0-4t2+4t2-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-4t2+4t2-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4+0-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-8t+8t-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4+0-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t2(t4-16)+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Apply the distributive property.
t2t4+t2⋅-16+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Multiply t2 by t4 by adding the exponents.
Use the power rule aman=am+n to combine exponents.
t2+4+t2⋅-16+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t6+t2⋅-16+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t6+t2⋅-16+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Move -16 to the left of t2.
t6-16t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
t6-16t2+3(t2+4)(t+2)(t-2)+7(4+t2)(2+t)(2-t)
Simplify terms.
Reorder terms.
t6-16t2+3(t2+4)(t+2)(t-2)+7(t2+4)(2+t)(2-t)
Reorder terms.
t6-16t2+3(t2+4)(t+2)(t-2)+7(t2+4)(t+2)(2-t)
Rewrite 2 as -1(-2).
t6-16t2+3(t2+4)(t+2)(t-2)+7(t2+4)(t+2)(-1(-2)-t)
Factor -1 out of -t.
t6-16t2+3(t2+4)(t+2)(t-2)+7(t2+4)(t+2)(-1(-2)-(t))
Factor -1 out of -1(-2)-(t).
t6-16t2+3(t2+4)(t+2)(t-2)+7(t2+4)(t+2)(-1(-2+t))
Simplify the expression.
Move a negative from the denominator of 7(t2+4)(t+2)(-1(-2+t)) to the numerator.
t6-16t2+3(t2+4)(t+2)(t-2)+-1⋅7(t2+4)(t+2)(-2+t)
Reorder terms.
t6-16t2+3(t2+4)(t+2)(t-2)+-1⋅7(t2+4)(t+2)(t-2)
t6-16t2+3(t2+4)(t+2)(t-2)+-1⋅7(t2+4)(t+2)(t-2)
Combine the numerators over the common denominator.
t6-16t2+3-1⋅7(t2+4)(t+2)(t-2)
t6-16t2+3-1⋅7(t2+4)(t+2)(t-2)
Simplify the numerator.
Multiply -1 by 7.
t6-16t2+3-7(t2+4)(t+2)(t-2)
Subtract 7 from 3.
t6-16t2-4(t2+4)(t+2)(t-2)
t6-16t2-4(t2+4)(t+2)(t-2)
Combine t^2+3/(t^4-16)+7/(16-t^4)

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