4-34. Review what you have learned in the previous lessons
by factoring the following expressions, if possible.
 81m2 – 1
• (9m + 1)(9m − 1)
 3n2 + 9n + 6
• (3n + 3)(n + 2)
or (n + 1)(3n + 6)
4.1.4 FACTORING
COMPLETELY
November 18, 2015
Objectives
• CO: SWBAT factor first a common
factor and then use the quadratic
factoring method.
• LO: SWBAT explain to a partner
how to factor completely.
4-35. Compare your factored expressions for problem 4-34 with the rest of your class.
a.
Is there more than one factored form of 3n2 + 9n + 6? Why or why not?
Yes, because you can build two different rectangles with algebra tiles.
b.
Why does 3n2 + 9n + 6 have more than one factored form while the other
quadratics in problem 4-34 only have one possible factored form? Look for
clues in the original expression and in the different factored forms.
Because it has a common 3 to be factored out of the expression.
c.
Without factoring, predict which quadratic expressions below may have
more than one factored form. Be prepared to defend your choice to the rest
of the class.
i.
12t2 – 10t + 2 ii.
5p2 – 23p + 10
i & iii; they have common factors.
iii.
10x2 + 25x − 15 iv.
3k2 + 7k – 6
4-36. FACTORING COMPLETELY
In part (c) of problem 4-35, you should have
noticed that each term in 12t2 – 10t + 2 is
divisible by 2. That is, each term in the
expression has a common factor of 2.
a. An expression is considered completely
factored if none of the factors can be
factored further. Often it is easiest to
remove common factors first, before
factoring with an area model. Rewrite the
expression 10x2 + 25x − 15 with the
common factor factored out.
5(2x2 + 5x − 3)
b.
Your result in part (a) is not completely
factored because the trinomial factor can
still be written as a product of two
factors. Factor 10x2 + 25x − 15 completely.
5(2x – 1)(x + 3)
6x
-3
2x2
-1x
-6x2
6x
-1x
5x
4-37. Factor each of the following expressions completely.
a. 5x2 + 15x − 20
5(x2 + 3x − 4)
4x + -1x
5(x + 4)(x − 1)
b. 3x3 − 6x2 – 45x
3x(x2 − 2x – 15)
-5x + 3x
3x(x + 3)(x − 5)
4-38. CLOSED SETS
Whole numbers (positive integers and zero) are said to be a closed set under
addition: if you add two whole numbers, you always get a whole number. Whole
numbers are not a closed set under subtraction: if you subtract two whole numbers,
you do not always get a whole number. For example, 2 – 5 = –3, and –3 is not a
whole number.
a. Investigate whether the integers are a closed set under addition, and whether
the integers are a closed set under subtraction. That is, will adding two
integers always give you an integer? Will subtracting two integers always give
you an integer? Give examples.
Integers are closed under addition: 1 + (-2) = -1
Integers are closed under subtraction: 3 – 4 = -1
b.
If you decided that integers are closed under either addition or subtraction, can
you explain how you know they are closed for all integers? Use mathematical
reasoning, not just examples, to justify your answer.
If you start at an integer on the number line and add or subtract an integer, you will always step
to the left or right along the integers; you never end up between integers.
c.
Now consider multiplication and division. Are integers a closed set under
multiplication? Are integers a closed set under division? Use examples and
reasoning to justify your answer.
Integers are closed under multiplication because if you multiply two integers you will never get a
fraction or decimal. (1 ∙ 3 = 3)
Integers are not closed under division, for example, 1 ÷ 2 = 1/2