Pre-Algebra Examples

Simplify 3(x+3)^2(2x-1)^-4-8(x+3)^3(2x-1)^-5
Step 1
Simplify each term.
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Step 1.1
Rewrite as .
Step 1.2
Expand using the FOIL Method.
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Step 1.2.1
Apply the distributive property.
Step 1.2.2
Apply the distributive property.
Step 1.2.3
Apply the distributive property.
Step 1.3
Simplify and combine like terms.
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Step 1.3.1
Simplify each term.
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Step 1.3.1.1
Multiply by .
Step 1.3.1.2
Move to the left of .
Step 1.3.1.3
Multiply by .
Step 1.3.2
Add and .
Step 1.4
Apply the distributive property.
Step 1.5
Simplify.
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Step 1.5.1
Multiply by .
Step 1.5.2
Multiply by .
Step 1.6
Rewrite the expression using the negative exponent rule .
Step 1.7
Multiply by .
Step 1.8
Simplify the numerator.
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Step 1.8.1
Factor out of .
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Step 1.8.1.1
Factor out of .
Step 1.8.1.2
Factor out of .
Step 1.8.1.3
Factor out of .
Step 1.8.1.4
Factor out of .
Step 1.8.1.5
Factor out of .
Step 1.8.2
Factor using the perfect square rule.
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Step 1.8.2.1
Rewrite as .
Step 1.8.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 1.8.2.3
Rewrite the polynomial.
Step 1.8.2.4
Factor using the perfect square trinomial rule , where and .
Step 1.9
Use the Binomial Theorem.
Step 1.10
Simplify each term.
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Step 1.10.1
Multiply by .
Step 1.10.2
Multiply by by adding the exponents.
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Step 1.10.2.1
Move .
Step 1.10.2.2
Multiply by .
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Step 1.10.2.2.1
Raise to the power of .
Step 1.10.2.2.2
Use the power rule to combine exponents.
Step 1.10.2.3
Add and .
Step 1.10.3
Raise to the power of .
Step 1.10.4
Raise to the power of .
Step 1.11
Apply the distributive property.
Step 1.12
Simplify.
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Step 1.12.1
Multiply by .
Step 1.12.2
Multiply by .
Step 1.12.3
Multiply by .
Step 1.13
Rewrite the expression using the negative exponent rule .
Step 1.14
Multiply by .
Step 1.15
Factor out of .
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Step 1.15.1
Factor out of .
Step 1.15.2
Factor out of .
Step 1.15.3
Factor out of .
Step 1.15.4
Factor out of .
Step 1.15.5
Factor out of .
Step 1.15.6
Factor out of .
Step 1.15.7
Factor out of .
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 by adding the exponents.
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Step 3.2.1
Multiply by .
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Step 3.2.1.1
Raise to the power of .
Step 3.2.1.2
Use the power rule to combine exponents.
Step 3.2.2
Add and .
Step 4
Combine the numerators over the common denominator.
Step 5
Simplify the numerator.
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Step 5.1
Rewrite as .
Step 5.2
Expand using the FOIL Method.
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Step 5.2.1
Apply the distributive property.
Step 5.2.2
Apply the distributive property.
Step 5.2.3
Apply the distributive property.
Step 5.3
Simplify and combine like terms.
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Step 5.3.1
Simplify each term.
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Step 5.3.1.1
Multiply by .
Step 5.3.1.2
Move to the left of .
Step 5.3.1.3
Multiply by .
Step 5.3.2
Add and .
Step 5.4
Apply the distributive property.
Step 5.5
Simplify.
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Step 5.5.1
Multiply by .
Step 5.5.2
Multiply by .
Step 5.6
Expand by multiplying each term in the first expression by each term in the second expression.
Step 5.7
Simplify each term.
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Step 5.7.1
Rewrite using the commutative property of multiplication.
Step 5.7.2
Multiply by by adding the exponents.
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Step 5.7.2.1
Move .
Step 5.7.2.2
Multiply by .
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Step 5.7.2.2.1
Raise to the power of .
Step 5.7.2.2.2
Use the power rule to combine exponents.
Step 5.7.2.3
Add and .
Step 5.7.3
Multiply by .
Step 5.7.4
Multiply by .
Step 5.7.5
Rewrite using the commutative property of multiplication.
Step 5.7.6
Multiply by by adding the exponents.
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Step 5.7.6.1
Move .
Step 5.7.6.2
Multiply by .
Step 5.7.7
Multiply by .
Step 5.7.8
Multiply by .
Step 5.7.9
Multiply by .
Step 5.7.10
Multiply by .
Step 5.8
Add and .
Step 5.9
Add and .
Step 5.10
Apply the distributive property.
Step 5.11
Simplify.
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Step 5.11.1
Multiply by .
Step 5.11.2
Multiply by .
Step 5.11.3
Multiply by .
Step 5.11.4
Multiply by .
Step 5.12
Subtract from .
Step 5.13
Subtract from .
Step 5.14
Subtract from .
Step 5.15
Subtract from .
Step 5.16
Rewrite in a factored form.
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Step 5.16.1
Factor using the rational roots test.
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Step 5.16.1.1
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 5.16.1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 5.16.1.3
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 5.16.1.3.1
Substitute into the polynomial.
Step 5.16.1.3.2
Raise to the power of .
Step 5.16.1.3.3
Multiply by .
Step 5.16.1.3.4
Raise to the power of .
Step 5.16.1.3.5
Multiply by .
Step 5.16.1.3.6
Subtract from .
Step 5.16.1.3.7
Multiply by .
Step 5.16.1.3.8
Add and .
Step 5.16.1.3.9
Subtract from .
Step 5.16.1.4
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 5.16.1.5
Divide by .
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Step 5.16.1.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+----
Step 5.16.1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
-
+----
Step 5.16.1.5.3
Multiply the new quotient term by the divisor.
-
+----
--
Step 5.16.1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
-
+----
++
Step 5.16.1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
+----
++
-
Step 5.16.1.5.6
Pull the next terms from the original dividend down into the current dividend.
-
+----
++
--
Step 5.16.1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
--
+----
++
--
Step 5.16.1.5.8
Multiply the new quotient term by the divisor.
--
+----
++
--
--
Step 5.16.1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
--
+----
++
--
++
Step 5.16.1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--
+----
++
--
++
-
Step 5.16.1.5.11
Pull the next terms from the original dividend down into the current dividend.
--
+----
++
--
++
--
Step 5.16.1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
---
+----
++
--
++
--
Step 5.16.1.5.13
Multiply the new quotient term by the divisor.
---
+----
++
--
++
--
--
Step 5.16.1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
---
+----
++
--
++
--
++
Step 5.16.1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
---
+----
++
--
++
--
++
Step 5.16.1.5.16
Since the remander is , the final answer is the quotient.
Step 5.16.1.6
Write as a set of factors.
Step 5.16.2
Factor by grouping.
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Step 5.16.2.1
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 5.16.2.1.1
Factor out of .
Step 5.16.2.1.2
Rewrite as plus
Step 5.16.2.1.3
Apply the distributive property.
Step 5.16.2.2
Factor out the greatest common factor from each group.
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Step 5.16.2.2.1
Group the first two terms and the last two terms.
Step 5.16.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.16.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.17
Combine exponents.
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Step 5.17.1
Factor out of .
Step 5.17.2
Rewrite as .
Step 5.17.3
Factor out of .
Step 5.17.4
Raise to the power of .
Step 5.17.5
Raise to the power of .
Step 5.17.6
Use the power rule to combine exponents.
Step 5.17.7
Add and .
Step 6
Move the negative in front of the fraction.