Basic Math Examples

Simplify ((3p^2-5p-2)/(y^2+y-2)*(y^2+5y-6)/(15p^2+8p+1))÷((7p^2-13p-2)/(10p^2-13p-3))
3p2-5p-2y2+y-2y2+5y-615p2+8p+1÷7p2-13p-210p2-13p-33p25p2y2+y2y2+5y615p2+8p+1÷7p213p210p213p3
Step 1
To divide by a fraction, multiply by its reciprocal.
3p2-5p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p25p2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2
Factor by grouping.
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Step 2.1
For a polynomial of the form ax2+bx+cax2+bx+c, rewrite the middle term as a sum of two terms whose product is ac=3-2=-6ac=32=6 and whose sum is b=-5b=5.
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Step 2.1.1
Factor -55 out of -5p5p.
3p2-5(p)-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p25(p)2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2.1.2
Rewrite -55 as 11 plus -66
3p2+(1-6)p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p2+(16)p2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2.1.3
Apply the distributive property.
3p2+1p-6p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p2+1p6p2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2.1.4
Multiply pp by 11.
3p2+p-6p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p2+p6p2y2+y2y2+5y615p2+8p+110p213p37p213p2
3p2+p-6p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-23p2+p6p2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2.2
Factor out the greatest common factor from each group.
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Step 2.2.1
Group the first two terms and the last two terms.
(3p2+p)-6p-2y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-2(3p2+p)6p2y2+y2y2+5y615p2+8p+110p213p37p213p2
Step 2.2.2
Factor out the greatest common factor (GCF) from each group.
p(3p+1)-2(3p+1)y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-2
p(3p+1)-2(3p+1)y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-2
Step 2.3
Factor the polynomial by factoring out the greatest common factor, 3p+1.
(3p+1)(p-2)y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-2
(3p+1)(p-2)y2+y-2y2+5y-615p2+8p+110p2-13p-37p2-13p-2
Step 3
Factor y2+y-2 using the AC method.
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Step 3.1
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 -2 and whose sum is 1.
-1,2
Step 3.2
Write the factored form using these integers.
(3p+1)(p-2)(y-1)(y+2)y2+5y-615p2+8p+110p2-13p-37p2-13p-2
(3p+1)(p-2)(y-1)(y+2)y2+5y-615p2+8p+110p2-13p-37p2-13p-2
Step 4
Factor y2+5y-6 using the AC method.
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Step 4.1
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 -6 and whose sum is 5.
-1,6
Step 4.2
Write the factored form using these integers.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+8p+110p2-13p-37p2-13p-2
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+8p+110p2-13p-37p2-13p-2
Step 5
Factor by grouping.
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Step 5.1
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is ac=151=15 and whose sum is b=8.
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Step 5.1.1
Factor 8 out of 8p.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+8(p)+110p2-13p-37p2-13p-2
Step 5.1.2
Rewrite 8 as 3 plus 5
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+(3+5)p+110p2-13p-37p2-13p-2
Step 5.1.3
Apply the distributive property.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+3p+5p+110p2-13p-37p2-13p-2
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)15p2+3p+5p+110p2-13p-37p2-13p-2
Step 5.2
Factor out the greatest common factor from each group.
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Step 5.2.1
Group the first two terms and the last two terms.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)(15p2+3p)+5p+110p2-13p-37p2-13p-2
Step 5.2.2
Factor out the greatest common factor (GCF) from each group.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)3p(5p+1)+1(5p+1)10p2-13p-37p2-13p-2
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)3p(5p+1)+1(5p+1)10p2-13p-37p2-13p-2
Step 5.3
Factor the polynomial by factoring out the greatest common factor, 5p+1.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)(5p+1)(3p+1)10p2-13p-37p2-13p-2
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)(5p+1)(3p+1)10p2-13p-37p2-13p-2
Step 6
Cancel the common factor of 3p+1.
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Step 6.1
Factor 3p+1 out of (5p+1)(3p+1).
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)(3p+1)(5p+1)10p2-13p-37p2-13p-2
Step 6.2
Cancel the common factor.
(3p+1)(p-2)(y-1)(y+2)(y-1)(y+6)(3p+1)(5p+1)10p2-13p-37p2-13p-2
Step 6.3
Rewrite the expression.
p-2(y-1)(y+2)(y-1)(y+6)5p+110p2-13p-37p2-13p-2
p-2(y-1)(y+2)(y-1)(y+6)5p+110p2-13p-37p2-13p-2
Step 7
Cancel the common factor of y-1.
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Step 7.1
Cancel the common factor.
p-2(y-1)(y+2)(y-1)(y+6)5p+110p2-13p-37p2-13p-2
Step 7.2
Rewrite the expression.
p-2y+2y+65p+110p2-13p-37p2-13p-2
p-2y+2y+65p+110p2-13p-37p2-13p-2
Step 8
Multiply p-2y+2 by y+65p+1.
(p-2)(y+6)(y+2)(5p+1)10p2-13p-37p2-13p-2
Step 9
Factor by grouping.
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Step 9.1
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is ac=10-3=-30 and whose sum is b=-13.
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Step 9.1.1
Factor -13 out of -13p.
(p-2)(y+6)(y+2)(5p+1)10p2-13(p)-37p2-13p-2
Step 9.1.2
Rewrite -13 as 2 plus -15
(p-2)(y+6)(y+2)(5p+1)10p2+(2-15)p-37p2-13p-2
Step 9.1.3
Apply the distributive property.
(p-2)(y+6)(y+2)(5p+1)10p2+2p-15p-37p2-13p-2
(p-2)(y+6)(y+2)(5p+1)10p2+2p-15p-37p2-13p-2
Step 9.2
Factor out the greatest common factor from each group.
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Step 9.2.1
Group the first two terms and the last two terms.
(p-2)(y+6)(y+2)(5p+1)(10p2+2p)-15p-37p2-13p-2
Step 9.2.2
Factor out the greatest common factor (GCF) from each group.
(p-2)(y+6)(y+2)(5p+1)2p(5p+1)-3(5p+1)7p2-13p-2
(p-2)(y+6)(y+2)(5p+1)2p(5p+1)-3(5p+1)7p2-13p-2
Step 9.3
Factor the polynomial by factoring out the greatest common factor, 5p+1.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2-13p-2
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2-13p-2
Step 10
Factor by grouping.
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Step 10.1
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is ac=7-2=-14 and whose sum is b=-13.
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Step 10.1.1
Factor -13 out of -13p.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2-13(p)-2
Step 10.1.2
Rewrite -13 as 1 plus -14
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2+(1-14)p-2
Step 10.1.3
Apply the distributive property.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2+1p-14p-2
Step 10.1.4
Multiply p by 1.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2+p-14p-2
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)7p2+p-14p-2
Step 10.2
Factor out the greatest common factor from each group.
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Step 10.2.1
Group the first two terms and the last two terms.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)(7p2+p)-14p-2
Step 10.2.2
Factor out the greatest common factor (GCF) from each group.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)p(7p+1)-2(7p+1)
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)p(7p+1)-2(7p+1)
Step 10.3
Factor the polynomial by factoring out the greatest common factor, 7p+1.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)(7p+1)(p-2)
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)(7p+1)(p-2)
Step 11
Cancel the common factor of p-2.
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Step 11.1
Factor p-2 out of (7p+1)(p-2).
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)(p-2)(7p+1)
Step 11.2
Cancel the common factor.
(p-2)(y+6)(y+2)(5p+1)(5p+1)(2p-3)(p-2)(7p+1)
Step 11.3
Rewrite the expression.
y+6(y+2)(5p+1)(5p+1)(2p-3)7p+1
y+6(y+2)(5p+1)(5p+1)(2p-3)7p+1
Step 12
Cancel the common factor of 5p+1.
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Step 12.1
Factor 5p+1 out of (y+2)(5p+1).
y+6(5p+1)(y+2)(5p+1)(2p-3)7p+1
Step 12.2
Cancel the common factor.
y+6(5p+1)(y+2)(5p+1)(2p-3)7p+1
Step 12.3
Rewrite the expression.
y+6y+22p-37p+1
y+6y+22p-37p+1
Step 13
Multiply y+6y+2 by 2p-37p+1.
(y+6)(2p-3)(y+2)(7p+1)
 [x2  12  π  xdx ]