Combine (y^2-13)/(y^2-2y-3)-1/(3-y)

Math
y2-13y2-2y-3-13-y
Factor y2-2y-3 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 -3 and whose sum is -2.
-3,1
Write the factored form using these integers.
y2-13(y-3)(y+1)-13-y
y2-13(y-3)(y+1)-13-y
Simplify with factoring out.
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Rewrite 3 as -1(-3).
y2-13(y-3)(y+1)-1-1(-3)-y
Factor -1 out of -y.
y2-13(y-3)(y+1)-1-1(-3)-(y)
Factor -1 out of -1(-3)-(y).
y2-13(y-3)(y+1)-1-1(-3+y)
Reorder terms.
y2-13(y-3)(y+1)-1-1(y-3)
y2-13(y-3)(y+1)-1-1(y-3)
To write y2-13(y-3)(y+1) as a fraction with a common denominator, multiply by -1-1.
y2-13(y-3)(y+1)⋅-1-1-1-1(y-3)
To write -1-1(y-3) as a fraction with a common denominator, multiply by y+1y+1.
y2-13(y-3)(y+1)⋅-1-1-1-1(y-3)⋅y+1y+1
Write each expression with a common denominator of (y-3)(y+1)⋅-1, by multiplying each by an appropriate factor of 1.
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Multiply y2-13(y-3)(y+1) and -1-1.
(y2-13)⋅-1(y-3)(y+1)⋅-1-1-1(y-3)⋅y+1y+1
Multiply 1-1(y-3) and y+1y+1.
(y2-13)⋅-1(y-3)(y+1)⋅-1-y+1-1(y-3)(y+1)
Reorder the factors of (y-3)(y+1)⋅-1.
(y2-13)⋅-1-(y+1)(y-3)-y+1-1(y-3)(y+1)
Reorder the factors of -1(y-3)(y+1).
(y2-13)⋅-1-(y+1)(y-3)-y+1-(y+1)(y-3)
(y2-13)⋅-1-(y+1)(y-3)-y+1-(y+1)(y-3)
Combine the numerators over the common denominator.
(y2-13)⋅-1-(y+1)-(y+1)(y-3)
Simplify the numerator.
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Apply the distributive property.
y2⋅-1-13⋅-1-(y+1)-(y+1)(y-3)
Move -1 to the left of y2.
-1⋅y2-13⋅-1-(y+1)-(y+1)(y-3)
Multiply -13 by -1.
-1⋅y2+13-(y+1)-(y+1)(y-3)
Rewrite -1y2 as -y2.
-y2+13-(y+1)-(y+1)(y-3)
Apply the distributive property.
-y2+13-y-1⋅1-(y+1)(y-3)
Multiply -1 by 1.
-y2+13-y-1-(y+1)(y-3)
Subtract 1 from 13.
-y2-y+12-(y+1)(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=-1⋅12=-12 and whose sum is b=-1.
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Factor -1 out of -y.
-y2-(y)+12-(y+1)(y-3)
Rewrite -1 as 3 plus -4
-y2+(3-4)y+12-(y+1)(y-3)
Apply the distributive property.
-y2+3y-4y+12-(y+1)(y-3)
-y2+3y-4y+12-(y+1)(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.
(-y2+3y)-4y+12-(y+1)(y-3)
Factor out the greatest common factor (GCF) from each group.
y(-y+3)+4(-y+3)-(y+1)(y-3)
y(-y+3)+4(-y+3)-(y+1)(y-3)
Factor the polynomial by factoring out the greatest common factor, -y+3.
(-y+3)(y+4)-(y+1)(y-3)
(-y+3)(y+4)-(y+1)(y-3)
(-y+3)(y+4)-(y+1)(y-3)
Reduce the expression by cancelling the common factors.
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Cancel the common factor of -y+3 and y-3.
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Factor -1 out of -y.
(-(y)+3)(y+4)-(y+1)(y-3)
Rewrite 3 as -1(-3).
(-(y)-1(-3))(y+4)-(y+1)(y-3)
Factor -1 out of -(y)-1(-3).
-(y-3)(y+4)-(y+1)(y-3)
Rewrite -(y-3) as -1(y-3).
-1(y-3)(y+4)-(y+1)(y-3)
Cancel the common factor.
-1(y-3)(y+4)-(y+1)(y-3)
Rewrite the expression.
-1(y+4)-(y+1)
-1(y+4)-(y+1)
Dividing two negative values results in a positive value.
y+4y+1
y+4y+1
Combine (y^2-13)/(y^2-2y-3)-1/(3-y)

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