Precalculus Examples

Factor 2x^3-9x^2-6x+5
Step 1
Factor using the rational roots test.
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Step 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 1.2
Find every combination of . These are the possible roots of the polynomial function.
Step 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 1.3.1
Substitute into the polynomial.
Step 1.3.2
Raise to the power of .
Step 1.3.3
Multiply by .
Step 1.3.4
Raise to the power of .
Step 1.3.5
Multiply by .
Step 1.3.6
Subtract from .
Step 1.3.7
Multiply by .
Step 1.3.8
Add and .
Step 1.3.9
Add and .
Step 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 1.5
Divide by .
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Step 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 1.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
+--+
Step 1.5.3
Multiply the new quotient term by the divisor.
+--+
++
Step 1.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
+--+
--
Step 1.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+--+
--
-
Step 1.5.6
Pull the next terms from the original dividend down into the current dividend.
+--+
--
--
Step 1.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
-
+--+
--
--
Step 1.5.8
Multiply the new quotient term by the divisor.
-
+--+
--
--
--
Step 1.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
-
+--+
--
--
++
Step 1.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
+--+
--
--
++
+
Step 1.5.11
Pull the next terms from the original dividend down into the current dividend.
-
+--+
--
--
++
++
Step 1.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
-+
+--+
--
--
++
++
Step 1.5.13
Multiply the new quotient term by the divisor.
-+
+--+
--
--
++
++
++
Step 1.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
-+
+--+
--
--
++
++
--
Step 1.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-+
+--+
--
--
++
++
--
Step 1.5.16
Since the remander is , the final answer is the quotient.
Step 1.6
Write as a set of factors.
Step 2
Factor by grouping.
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Step 2.1
Factor by grouping.
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Step 2.1.1
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 2.1.1.1
Factor out of .
Step 2.1.1.2
Rewrite as plus
Step 2.1.1.3
Apply the distributive property.
Step 2.1.2
Factor out the greatest common factor from each group.
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Step 2.1.2.1
Group the first two terms and the last two terms.
Step 2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 2.2
Remove unnecessary parentheses.