Assignment
Notes
For Monday:
1. Exercises from 5.4, 5.5
2. Exam 2: October 30 - Chapter 4 & 5, and Cumulative
Portion
Study guide: Chapter 4 Test, p 294, # 1 - 5. Chapter 5
Test, p 367 # 1 - 19. Cumulative: Chapter 1 - 4, p 295,
# 1 - 8, 11 - 20.
(Math 1010)
M 1010 §5.4, 5.5
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§5.4 - Factoring
Notes
Factoring polynomials is the reverse process of
multiplication of polynomials. The Distributive Property is
used.
Example: Multiplying: 4x(3x 2 + 7) = 12x 3 + 28x
Factoring: 12x 3 + 28x = 4x(3x 2 + 7)
The greatest common monomial factor of terms in a
polynomial is the greatest integer a dividing all
coefficients with the highest-powered variable factor x n in
common with all terms.
In the above example this is the factor 4x.
Example: What is the GCMF of 24x 3 − 32x 2 ? Answer:
8x 2 .
(Math 1010)
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§5.4 - Factoring
Notes
Examples: Factor out the greatest common monomial.
1. −3x 2 + 12x − 18
2. −6x 5 + 30x 4 − 12x 3
3. 12x 2 y + 28xy 2
(Math 1010)
M 1010 §5.4, 5.5
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§5.4 - Factoring
Notes
Factoring out the Greatest Common Polynomial
sometimes involve factoring out a binomial. This is called
factoring by grouping.
Example x(7x + 2) + 4(7x + 2) = (7x + 2)(x + 4)
Tip: Rewrite polynomials in standard form before trying
to factor by grouping. Check to see if a monomial factors
out of the first two terms, and repeat for each other pair
of terms. Groups do not always appear in order.
Example
x 3 − 5x 2 + x − 5 = x 2 (x − 5) + 1(x − 5) = (x − 5)(x 2 + 1).
Try 4x 3 − 8x 2 + 3x − 6.
(Math 1010)
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§5.4 - Factoring
Notes
Special factoring exists as the difference of two squares.
Example: a2 − b 2 = (a + b)(a − b)
1. x 2 − 64
2. (x + 3)2 − 4
Formulas exists for the difference of two cubes (a3 − b 3 )
and the sum of two cubes (a3 + b 3 ).
(Math 1010)
M 1010 §5.4, 5.5
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§5.4 - Factoring
Notes
Exercises:
# 65 Factor completely: x 2 + 25x + x + 25
# 77 x 2 − 9
# 93 (x − 1)2 − 16
# 115 y 4 − 81
(Math 1010)
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§5.5 - Factoring Trinomials
Notes
Factoring a trinomial ax 2 + bx + c like x 2 + 6x + 9,
4x 2 + 5x − 6, or x 2 + 26x + 25 uses a guess and check
method. A complete list of all factors can be used as a
test, and each one can be multiplied to see if it works.
(Math 1010)
M 1010 §5.4, 5.5
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§5.5 - Factoring Trinomials
Notes
Example: Factor x 2 − 17x − 18. Tips: The sign of −18
is negative, so its factors must differ in sign. | − 17| is
close to | − 18|; first guess factors of -18 that are far
apart (in absolute value).
−18 has factors −1, 18 and 1, −18 that are far apart.
The factoring is (x + 1)(x − 18). The check is
(x + 1)(x − 18) = x 2 − 18x + x − 18 = x 2 − 17x − 18.
(Math 1010)
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§5.5 - Factoring Trinomials
Notes
Exercises:
Example 2a: Factor x 2 − 2x − 8
Example 7: Factor 2x 2 − x − 21
# 37: Factor x 2 + 6x + 5
# 41: Factor y 2 + 7y − 30
# 67: Factor 6x 2 − 5x − 25
# 87: Factor 60y 3 + 35y 2 − 50y
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