Any Polynomial—Look for the Greatest Common Factor :
xy - xz = x(y - z)
Example :
6a^{2}b + 10ab^{2} = 2ab(3a + 5b)
Binomials—Look for a Difference of Two Squares :
x^{2} - y^{2} = (x + y)(x - y)
Example :
a^{2} - 9b^{2} = (a + 3b)(a - 3b)
Trinomials—Look for Perfect-Square Trinomials :
x^{2} + 2xy + y^{2} = (x + y)^{2}
x^{2} - 2xy + y^{2} = (x - y)^{2}
Examples :
a^{2} + 4a + 4 = (a + 2)^{2}
a^{2} - 2a + 1 = (a - 1)^{2}
Other Factorable Trinomials :
x^{2} + bx + c = (x + _ ) (x + _ )
ax^{2} + bx + c = ( _ x + _ ) ( _ x + _ )
Examples :
y^{2} + 3y + 2 = (y + 1)(y + 2)
6y^{2} + 7y + 2 = (2y + 1)(3y + 2)
Polynomials of Four or More Terms - Factor by grouping :
ax + bx + ay + by :
= x(a + b) + y(a + b)
= (x + y)(a + b)
Example :
2y^{3} + 4y^{2} + y + 2 :
= (2y^{3} + 4y^{2}) + (y + 2)
= 2y^{2}(y + 2) + 1(y + 2)
= (y + 2)(2y^{2} + 1)
Note :
If none of the factoring methods work, the polynomial is unfactorable.
Remember :
For a polynomial of the form ax^{2} + bx + c, if there are no integers whose sum is b and whose product is ac, then the polynomial is unfactorable.
Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.
To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.
Step 1 :
Check for a greatest common factor.
Step 2 :
Check for a pattern that fits the difference of two squares or a perfect-square trinomial.
Step 3 :
To factor x^{2} + bx + c, look for two numbers whose sum is b and whose product is c.
To factor ax^{2} + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.
Step 4 :
Check for common factors.
Tell whether each expression is completely factored. If not, factor it.
Example 1 :
2a(a^{2} + 4)
Neither 2a nor a^{2} + 4 can be factored further.
2a(a^{2} + 4) is completely factored.
Example 2 :
(2a + 6)(a + 5)
2a + 6 can be further factored.
Factor out 2, the GCF of 2a and 6.
= 2(a + 3)(a + 5)
2(a + 3)(a + 5) is completely factored.
Example 3 :
Factor -2ab^{2} + 16ab - 32a completely. Check your answer.
= -2ab^{2} + 16ab - 32a
Factor out the GCF.
= -2a(b^{2} - 8b + 16)
b^{2} + 8b + 16 is a perfect square trinomial of the form
x^{2} + 2xy + y^{2}
x = b and y = 4.
= -2a(b - 4)^{2}
Check :
-2a(b - 4)^{2 }= -2a(b^{2} - 8b + 16)
= -2ab^{2} + 16ab - 32a ✓
Factor each polynomial completely.
Example 4 :
2x^{2} + 5x + 4
The GCF is 1 and there is no pattern.
= ( _ x + _ ) ( _ x + _ )
a = 2 and c = 4; Outer + Inner = 5.
2x^{2} + 5x + 4 is unfactorable.
Example 5 :
3m^{4} - 15m^{3} + 12m^{2}
Factor out the GCF.
= 3m^{2}(m^{2} - 5m + 4)
There is no pattern.
= 3m^{2}(m + _ )(m + _ )
b = -5 and c = 4; look for factors of 4 whose sum is -5.
Factors of 4 -1 and -4 |
Sum -5 ✓ |
The factors needed are -1 and -4.
= 3m^{2}(m - 1)(m - 4)
Example 6 :
4y^{3} + 18y^{2} + 20y
Factor out the GCF.
= 2y(2y^{2} + 9y + 10)
There is no pattern.
= 2y( _ y + _ )( _ y + _ )
a = 2 and c = 10; Outer + Inner = 9
= 2y(y + 2)(2y + 5)
Example 7 :
n^{5} - n
Factor out the GCF.
= n(n^{4} - 1)
n^{4} - 1 is a difference of two squares.
= n(n^{2} + 1)(n^{2} - 1)
n^{2} - 1 is a difference of two squares.
= n(n^{2} + 1)(n + 1)(n - 1)
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