Basic Math Examples

Solve for y (1-y)((2-y)(-3-y))-24+12y=0
Step 1
Simplify .
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Step 1.1
Simplify each term.
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Step 1.1.1
Expand using the FOIL Method.
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Step 1.1.1.1
Apply the distributive property.
Step 1.1.1.2
Apply the distributive property.
Step 1.1.1.3
Apply the distributive property.
Step 1.1.2
Simplify and combine like terms.
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Step 1.1.2.1
Simplify each term.
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Step 1.1.2.1.1
Multiply by .
Step 1.1.2.1.2
Multiply by .
Step 1.1.2.1.3
Multiply by .
Step 1.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.1.2.1.5
Multiply by by adding the exponents.
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Step 1.1.2.1.5.1
Move .
Step 1.1.2.1.5.2
Multiply by .
Step 1.1.2.1.6
Multiply by .
Step 1.1.2.1.7
Multiply by .
Step 1.1.2.2
Subtract from .
Step 1.1.3
Expand by multiplying each term in the first expression by each term in the second expression.
Step 1.1.4
Simplify each term.
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Step 1.1.4.1
Multiply by .
Step 1.1.4.2
Multiply by .
Step 1.1.4.3
Multiply by .
Step 1.1.4.4
Rewrite using the commutative property of multiplication.
Step 1.1.4.5
Multiply by by adding the exponents.
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Step 1.1.4.5.1
Move .
Step 1.1.4.5.2
Multiply by .
Step 1.1.4.6
Multiply by .
Step 1.1.4.7
Move to the left of .
Step 1.1.4.8
Rewrite using the commutative property of multiplication.
Step 1.1.4.9
Multiply by by adding the exponents.
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Step 1.1.4.9.1
Move .
Step 1.1.4.9.2
Multiply by .
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Step 1.1.4.9.2.1
Raise to the power of .
Step 1.1.4.9.2.2
Use the power rule to combine exponents.
Step 1.1.4.9.3
Add and .
Step 1.1.5
Combine the opposite terms in .
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Step 1.1.5.1
Subtract from .
Step 1.1.5.2
Add and .
Step 1.1.6
Add and .
Step 1.2
Simplify by adding terms.
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Step 1.2.1
Subtract from .
Step 1.2.2
Add and .
Step 2
Factor the left side of the equation.
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Step 2.1
Factor out of .
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Step 2.1.1
Reorder and .
Step 2.1.2
Factor out of .
Step 2.1.3
Factor out of .
Step 2.1.4
Rewrite as .
Step 2.1.5
Factor out of .
Step 2.1.6
Factor out of .
Step 2.2
Factor using the rational roots test.
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Step 2.2.1
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 2.2.2
Find every combination of . These are the possible roots of the polynomial function.
Step 2.2.3
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 2.2.3.1
Substitute into the polynomial.
Step 2.2.3.2
Raise to the power of .
Step 2.2.3.3
Multiply by .
Step 2.2.3.4
Subtract from .
Step 2.2.3.5
Add and .
Step 2.2.4
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 2.2.5
Divide by .
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Step 2.2.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Step 2.2.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 2.2.5.3
Multiply the new quotient term by the divisor.
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+-
Step 2.2.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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-+
Step 2.2.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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+
Step 2.2.5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 2.2.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
+
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Step 2.2.5.8
Multiply the new quotient term by the divisor.
+
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Step 2.2.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
+
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-+
Step 2.2.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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-
Step 2.2.5.11
Pull the next terms from the original dividend down into the current dividend.
+
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-+
Step 2.2.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
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-+
Step 2.2.5.13
Multiply the new quotient term by the divisor.
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-+
Step 2.2.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 2.2.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 2.2.5.16
Since the remander is , the final answer is the quotient.
Step 2.2.6
Write as a set of factors.
Step 2.3
Factor.
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Step 2.3.1
Factor using the AC method.
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Step 2.3.1.1
Factor using the AC method.
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Step 2.3.1.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.3.1.1.2
Write the factored form using these integers.
Step 2.3.1.2
Remove unnecessary parentheses.
Step 2.3.2
Remove unnecessary parentheses.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Add to both sides of the equation.
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Add to both sides of the equation.
Step 6
Set equal to and solve for .
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Step 6.1
Set equal to .
Step 6.2
Subtract from both sides of the equation.
Step 7
The final solution is all the values that make true.