Basic Math Examples

Simplify ((5a^2-a)/(25a^2-10a+1)+4/(1-25a^2))÷(1-3/(5a-1))-a/(5a+1)
Step 1
To write as a fraction with a common denominator, multiply by .
Step 2
To write as a fraction with a common denominator, multiply by .
Step 3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 3.1
Multiply by .
Step 3.2
Multiply by .
Step 3.3
Reorder the factors of .
Step 4
Combine the numerators over the common denominator.
Step 5
Simplify each term.
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Step 5.1
Rewrite the division as a fraction.
Step 5.2
Multiply the numerator by the reciprocal of the denominator.
Step 5.3
Simplify the numerator.
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Step 5.3.1
Expand using the FOIL Method.
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Step 5.3.1.1
Apply the distributive property.
Step 5.3.1.2
Apply the distributive property.
Step 5.3.1.3
Apply the distributive property.
Step 5.3.2
Simplify each term.
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Step 5.3.2.1
Multiply by .
Step 5.3.2.2
Rewrite using the commutative property of multiplication.
Step 5.3.2.3
Multiply by by adding the exponents.
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Step 5.3.2.3.1
Move .
Step 5.3.2.3.2
Use the power rule to combine exponents.
Step 5.3.2.3.3
Add and .
Step 5.3.2.4
Multiply by .
Step 5.3.2.5
Multiply by .
Step 5.3.2.6
Rewrite using the commutative property of multiplication.
Step 5.3.2.7
Multiply by by adding the exponents.
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Step 5.3.2.7.1
Move .
Step 5.3.2.7.2
Multiply by .
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Step 5.3.2.7.2.1
Raise to the power of .
Step 5.3.2.7.2.2
Use the power rule to combine exponents.
Step 5.3.2.7.3
Add and .
Step 5.3.2.8
Multiply by .
Step 5.3.3
Apply the distributive property.
Step 5.3.4
Simplify.
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Step 5.3.4.1
Multiply by .
Step 5.3.4.2
Multiply by .
Step 5.3.4.3
Multiply by .
Step 5.3.5
Add and .
Step 5.3.6
Subtract from .
Step 5.4
Simplify the denominator.
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Step 5.4.1
Rewrite as .
Step 5.4.2
Rewrite as .
Step 5.4.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 5.4.4
Multiply by .
Step 5.4.5
Factor using the perfect square rule.
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Step 5.4.5.1
Rewrite as .
Step 5.4.5.2
Rewrite as .
Step 5.4.5.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 5.4.5.4
Rewrite the polynomial.
Step 5.4.5.5
Factor using the perfect square trinomial rule , where and .
Step 5.4.6
Combine exponents.
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Step 5.4.6.1
Rewrite as .
Step 5.4.6.2
Factor out of .
Step 5.4.6.3
Factor out of .
Step 5.4.6.4
Reorder terms.
Step 5.4.6.5
Multiply by by adding the exponents.
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Step 5.4.6.5.1
Move .
Step 5.4.6.5.2
Multiply by .
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Step 5.4.6.5.2.1
Raise to the power of .
Step 5.4.6.5.2.2
Use the power rule to combine exponents.
Step 5.4.6.5.3
Add and .
Step 5.4.7
Factor out negative.
Step 5.5
Move the negative in front of the fraction.
Step 5.6
Simplify the denominator.
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Step 5.6.1
Write as a fraction with a common denominator.
Step 5.6.2
Combine the numerators over the common denominator.
Step 5.6.3
Subtract from .
Step 5.7
Multiply the numerator by the reciprocal of the denominator.
Step 5.8
Multiply by .
Step 5.9
Cancel the common factor of .
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Step 5.9.1
Move the leading negative in into the numerator.
Step 5.9.2
Factor out of .
Step 5.9.3
Cancel the common factor.
Step 5.9.4
Rewrite the expression.
Step 5.10
Multiply by .
Step 5.11
Move the negative in front of the fraction.
Step 6
Reorder terms.
Step 7
To write as a fraction with a common denominator, multiply by .
Step 8
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 8.1
Multiply by .
Step 8.2
Reorder the factors of .
Step 8.3
Reorder the factors of .
Step 9
Combine the numerators over the common denominator.
Step 10
Simplify the numerator.
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Step 10.1
Apply the distributive property.
Step 10.2
Simplify.
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Step 10.2.1
Multiply by .
Step 10.2.2
Multiply by .
Step 10.2.3
Multiply by .
Step 10.2.4
Multiply by .
Step 10.2.5
Multiply by .
Step 10.3
Rewrite as .
Step 10.4
Expand using the FOIL Method.
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Step 10.4.1
Apply the distributive property.
Step 10.4.2
Apply the distributive property.
Step 10.4.3
Apply the distributive property.
Step 10.5
Simplify and combine like terms.
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Step 10.5.1
Simplify each term.
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Step 10.5.1.1
Rewrite using the commutative property of multiplication.
Step 10.5.1.2
Multiply by by adding the exponents.
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Step 10.5.1.2.1
Move .
Step 10.5.1.2.2
Multiply by .
Step 10.5.1.3
Multiply by .
Step 10.5.1.4
Multiply by .
Step 10.5.1.5
Multiply by .
Step 10.5.1.6
Multiply by .
Step 10.5.2
Subtract from .
Step 10.6
Expand by multiplying each term in the first expression by each term in the second expression.
Step 10.7
Simplify each term.
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Step 10.7.1
Rewrite using the commutative property of multiplication.
Step 10.7.2
Multiply by by adding the exponents.
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Step 10.7.2.1
Move .
Step 10.7.2.2
Multiply by .
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Step 10.7.2.2.1
Raise to the power of .
Step 10.7.2.2.2
Use the power rule to combine exponents.
Step 10.7.2.3
Add and .
Step 10.7.3
Multiply by .
Step 10.7.4
Rewrite using the commutative property of multiplication.
Step 10.7.5
Multiply by by adding the exponents.
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Step 10.7.5.1
Move .
Step 10.7.5.2
Multiply by .
Step 10.7.6
Multiply by .
Step 10.7.7
Multiply by .
Step 10.7.8
Multiply by .
Step 10.7.9
Multiply by .
Step 10.7.10
Multiply by .
Step 10.8
Subtract from .
Step 10.9
Add and .
Step 10.10
Apply the distributive property.
Step 10.11
Simplify.
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Step 10.11.1
Rewrite using the commutative property of multiplication.
Step 10.11.2
Rewrite using the commutative property of multiplication.
Step 10.11.3
Rewrite using the commutative property of multiplication.
Step 10.11.4
Multiply by .
Step 10.12
Simplify each term.
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Step 10.12.1
Multiply by by adding the exponents.
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Step 10.12.1.1
Move .
Step 10.12.1.2
Multiply by .
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Step 10.12.1.2.1
Raise to the power of .
Step 10.12.1.2.2
Use the power rule to combine exponents.
Step 10.12.1.3
Add and .
Step 10.12.2
Multiply by .
Step 10.12.3
Multiply by by adding the exponents.
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Step 10.12.3.1
Move .
Step 10.12.3.2
Multiply by .
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Step 10.12.3.2.1
Raise to the power of .
Step 10.12.3.2.2
Use the power rule to combine exponents.
Step 10.12.3.3
Add and .
Step 10.12.4
Multiply by .
Step 10.12.5
Multiply by by adding the exponents.
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Step 10.12.5.1
Move .
Step 10.12.5.2
Multiply by .
Step 10.12.6
Multiply by .
Step 10.13
Subtract from .
Step 10.14
Add and .
Step 10.15
Add and .
Step 10.16
Subtract from .
Step 10.17
Add and .
Step 10.18
Rewrite in a factored form.
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Step 10.18.1
Factor using the rational roots test.
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Step 10.18.1.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 10.18.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 10.18.1.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 10.18.1.3.1
Substitute into the polynomial.
Step 10.18.1.3.2
Raise to the power of .
Step 10.18.1.3.3
Multiply by .
Step 10.18.1.3.4
Raise to the power of .
Step 10.18.1.3.5
Multiply by .
Step 10.18.1.3.6
Subtract from .
Step 10.18.1.3.7
Multiply by .
Step 10.18.1.3.8
Add and .
Step 10.18.1.3.9
Subtract from .
Step 10.18.1.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 10.18.1.5
Divide by .
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Step 10.18.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
--+-
Step 10.18.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
--+-
Step 10.18.1.5.3
Multiply the new quotient term by the divisor.
--+-
+-
Step 10.18.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
--+-
-+
Step 10.18.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--+-
-+
-
Step 10.18.1.5.6
Pull the next terms from the original dividend down into the current dividend.
--+-
-+
-+
Step 10.18.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
-
--+-
-+
-+
Step 10.18.1.5.8
Multiply the new quotient term by the divisor.
-
--+-
-+
-+
-+
Step 10.18.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
-
--+-
-+
-+
+-
Step 10.18.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
--+-
-+
-+
+-
+
Step 10.18.1.5.11
Pull the next terms from the original dividend down into the current dividend.
-
--+-
-+
-+
+-
+-
Step 10.18.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
-+
--+-
-+
-+
+-
+-
Step 10.18.1.5.13
Multiply the new quotient term by the divisor.
-+
--+-
-+
-+
+-
+-
+-
Step 10.18.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
-+
--+-
-+
-+
+-
+-
-+
Step 10.18.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-+
--+-
-+
-+
+-
+-
-+
Step 10.18.1.5.16
Since the remander is , the final answer is the quotient.
Step 10.18.1.6
Write as a set of factors.
Step 10.18.2
Factor by grouping.
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Step 10.18.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 10.18.2.1.1
Factor out of .
Step 10.18.2.1.2
Rewrite as plus
Step 10.18.2.1.3
Apply the distributive property.
Step 10.18.2.2
Factor out the greatest common factor from each group.
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Step 10.18.2.2.1
Group the first two terms and the last two terms.
Step 10.18.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 10.18.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 10.18.3
Combine like factors.
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Step 10.18.3.1
Raise to the power of .
Step 10.18.3.2
Raise to the power of .
Step 10.18.3.3
Use the power rule to combine exponents.
Step 10.18.3.4
Add and .
Step 11
Reduce the expression by cancelling the common factors.
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Step 11.1
Cancel the common factor of .
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Step 11.1.1
Cancel the common factor.
Step 11.1.2
Rewrite the expression.
Step 11.2
Cancel the common factor of .
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Step 11.2.1
Cancel the common factor.
Step 11.2.2
Rewrite the expression.