1. Factoring Learning Resource Services
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Factoring

2. Polynomials Examples:
A term is the product of a number and/or a variable raised to a non-negative integer power, 3x4y, for example. A polynomial is an expression consisting of one or more terms.
Examples:
A monomial has one term
A binomial has two terms x xy
A trinomial has three terms p q + r
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3. Degree of a Polynomial
The single largest total number of degrees (or powers) to which the variables in any one term are raised is called the degree of a polynomial.
Examples:
2x3 degree = 3 variable x raised to the third power
4x2y1 degree = 3 variable x raised to the second power plus variable y raised to the first power
x4y3 - 4x2y1 degree = 7 variable x raised to the fourth power plus variable y raised to the third power Note: the second term, 4x2y1, has degree = 3, since variable x is raised to the second power and variable y is raised to the first power
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Factoring

4. Factoring a Polynomial
Remove common factors
Express the polynomial as a product of its factors
Examples:
Polynomial Factored Form
2x3 + 8x x2 (x + 4)
3z5 + 3z4 - 12z (z5 + z4 - 4z - 4)
3(z + 1)(z2 + 2)(z2 - 2)
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5. Removal of Common Factors
For 6x2 + 12x - 3
Remove the common factor of 3: (2x2 + 4x - 1)
For x2y3z6 - x4y3z4 + x2y5z4
Remove common factor of x2y3z4:
x2y3z4(z2 - x2 + y2)
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6. Factoring Binomials Difference of Squares Difference of Cubes
Sum of Cubes
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7. Factoring Binomials: Difference of Squares
A2 - B2 = (A - B)(A + B)
Examples:
x2 - y2 = (x - y)(x + y)
4r2 - 9s2 = (2r)2 - (3s)2
= (2r - 3s)(2r + 3s)
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8. Factoring Binomials: Difference of Cubes
A3 - B3 = (A - B)(A2 + AB + B2)
Examples:
x3 - y3 = (x - y)(x2 + xy + y2) cannot be further factored
2z = 2(z6 - 27)
= 2[(z2)3 - 33]
= 2[(z2 - 3)(z4 + 3z2  32)]
= 2(z2 - 3)(z4 + 3z2 + 9)
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9. Factoring Binomials: Sum of Cubes
A3 + B3 = (A + B)(A2 - AB + B2)
Examples:
x3 + y3 = (x + y)(x2 - xy + y2)
x9 + 1 = (x3)3 + 13
= (x3 + 1)[(x3)2 - x3 + 1)]
= (x3 + 1)(x6 - x3 + 1)
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Factoring

10. Factoring Trinomials For the common form of the trinomial ax2 + bx + c
let a1 and a2 be factors of a
let s1 and s2 be factors of c
let b be expressed as a1s2 + a2s1
Then the factored form of the trinomial becomes
(a1x + s1)(a2x + s2)
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Factoring

11. Factoring Monic Quadratic Trinomials
In the monic form of the trinomial ax2 + bx + c, the
value of a is one. The factored form is represented by
(x + s1)(x + s2).
a1a2 = 1, s1 + s2 = b, s1s2 = c
Example:
For the trinomial x2 + 3x a = 1, b = 3, c = -40
s1 + s2 = 3
s1s2 = -40
The values for s1 and s2 are found by trial and error …
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Factoring

12. Finding s1 and s2 Example:
For the trinomial x2 + 3x a = 1, b = 3, c = -40
The factors of -40 are
-1 and and -40
-2 and and -20
-4 and and -10
-8 and 5 8 and -5
Test each pair of factors. One set will sum to 3. Use
that set to substitute for s1 and s2 in the factored form
for the trinomial.
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13. Finding s1 and s2 (cont) Example:
For the trinomial x2 + 3x a = 1, b = 3, c = -40
The pair of factors which sum to 3 are 8 and -5.
The factored form for the trinomial becomes (x + 8)(x - 5)
Multiply out to check:
x2 - 5x + 8x - 40
x2 + 3x - 40
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14. Monic Quadratics Not every monic quadratic trinomial can be factored
You may need to use the quadratic formula
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15. Factoring Nonmonic Quadratic Trinomials ax2 + bx + c
If possible, express ac as a product of two integers
that have a sum of b.
Example:
For the trinomial 6x2 - 19x a = 6, b = -19, c = 15
and ac = 90
By trial and error, find two integers
whose product = 90 and sum = -19 …
... the two integers are -9 and -10
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16. Factoring ax2 + bx + c (cont)
Example:
For the trinomial 6x2 - 19x a = 6, b = -19, c = 15
and ac = 90
Rewrite the term bx using the two integers -9 and x2 + (-9x -10x) + 15
Group the 4 terms into 2 pairs
(6x2 - 9x) + (-10x + 15)
Remove the common factors from each pair
3x(2x - 3) - 5(2x - 3)
Distribute the term that is a common factor
(2x - 3)(3x - 5)
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17. Factoring Polynomials With Four Terms by Grouping
Example:
Factor the polynomial x3 + x2 - x - 1
Group into pairs and remove common factors
(x3 + x2) + (-x - 1)
x2(x + 1) + (-1)(x + 1)
x2(x + 1) - 1(x + 1)
Distribute the term that is a common factor
(x + 1)(x2 - 1)
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18. Algebra Review Homework 1
Example:
x4y + 27xy4
common factor xy
xy (x3 + 27y3)
(x3 + 27y3) is a sum of cubes
sum of cubes where A = x and B = 3y
xy [x3 + (3y)3]
xy (x + 3y) (x2 - 3xy + 9y2 )
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19. Algebra Review Homework 2
Example:
q6 - 1
difference of squares
( q3)2 - 12
(q3 - 1) (q3 + 1)
(q3 - 1) difference of cubes (q3 + 1) sum of cubes
(q - 1) (q2 + q + 1) (q + 1) (q2 - q + 1)
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Factoring