MATH 130 · MONDAY, AUGUST 31
Split the middle term.
Factor by grouping.
For ax² + bx + c: product ac, sum b.
Then split, group, factor, and check.
Find a product-and-sum pair
E1 · ENTRANCE · TOP HAT
Product = 18
Sum = 11
Which pair satisfies
both conditions?
A.
6 and 3
B.
12 and −1
C.
9 and 2
D.
−9 and −2
C · 9 and 2: product 18, sum 11.
Reverse the distributive property
E2 · ENTRANCE · TOP HAT
3x(2x + 3) + (2x + 3)
Factor out the
common binomial.
A.
(3x+1)(2x+3)
B.
3x(2x+3)
C.
4x(2x+3)
D.
(3x)(1)(2x+3)
A · (3x+1)(2x+3). The last binomial has coefficient 1.
One method, in this order
THE ROUTINE
0 Take out any GCF.
1 Find m and n: mn = ac, m + n = b.
2 Replace bx with mx + nx.
3 Group in pairs; factor each pair.
4 Factor out the common binomial.
5 Expand to check.
ax² + bx + c
Find the pair before you split
W1 · WORKED EXAMPLE
6x² + 11x + 3
ac = 6 · 3 = 18
m + n = 11
Use 9 and 2.
Factor pairs of 18
1 and 18 → sum 19
2 and 9 → sum 11
3 and 6 → sum 9
Split one term into two
W1 · SPLIT → GROUP
6x² + 11x + 3
= 6x² + 9x + 2x + 3
= (6x² + 9x) + (2x + 3)
= 3x(2x + 3) + 1(2x + 3)
Same expression
9x + 2x = 11x
The two groups share
the binomial 2x + 3.
The common factor is a binomial
W1 · FACTOR → CHECK
3x(2x + 3) + 1(2x + 3)
= (3x + 1)(2x + 3)
Check: 6x² + 9x + 2x + 3 = 6x² + 11x + 3
Your turn: show the middle line
P1 · GUIDED PRACTICE
6x² + 7x + 2
Show these steps
ac and the pair
The four-term line
Two factored groups
The final product
Use product ac and sum b.
Then split, group, and factor.
Check the grouping, not just the answer
P1 · CHECK
ac = 12; 3 + 4 = 7
6x² + 3x + 4x + 2
= 3x(2x + 1) + 2(2x + 1)
= (3x + 2)(2x + 1)
Which split keeps both conditions?
H1 · CHECKPOINT · TOP HAT
6x² + x − 2
Product = −12
Sum = 1
A.
6x² + 3x − 2x − 2
B.
6x² + 4x − 3x − 2
C.
6x² − 4x + 3x − 2
D.
6x² + 6x − 5x − 2
B · 4 and −3: product −12, sum 1.
A negative group needs care
W2 · SPLIT → GROUP
6x² + x − 2
= 6x² + 4x − 3x − 2
= (6x² + 4x) + (−3x − 2)
= 2x(3x + 2) − 1(3x + 2)
−1(3x + 2) = −3x − 2
Match the entire binomial
W2 · FACTOR → CHECK
2x(3x + 2) − 1(3x + 2)
= (2x − 1)(3x + 2)
Check: 6x² + 4x − 3x − 2 = 6x² + x − 2
Take out the GCF before ac
W3 · GCF FIRST
6x² + 15x + 9
= 3(2x² + 5x + 3)
= 3[2x² + 2x + 3x + 3]
= 3[2x(x + 1) + 3(x + 1)]
= 3(2x + 3)(x + 1)
Explain every factoring step
P2 · PARTNER PRACTICE
6x² − 15x + 9
Take out the GCF first.
Use the quadratic that remains to find ac.
Split, group, factor, and expand to check.
Keep the GCF and the negative signs
P2 · CHECK
6x² − 15x + 9 = 3(2x² − 5x + 3)
= 3[2x² − 2x − 3x + 3]
= 3[2x(x − 1) − 3(x − 1)]
= 3(2x − 3)(x − 1)
Do the full method independently
P3 · INDEPENDENT PRACTICE
4x² − 4x − 3
Show the pair, the split, both groups,
and the factored product.
Expand to check your work.
Check each link in the method
P3 · CHECK
4x² − 4x − 3
= 4x² − 6x + 2x − 3
= 2x(2x − 3) + 1(2x − 3)
= (2x + 1)(2x − 3)
Show that you can use the method
X1 · WRITTEN EXIT CHECK
6x² − 7x − 3
Factor by splitting the middle term
and then grouping. Show every step.
Expand to check.
6x²−9x+2x−3 = 3x(2x−3)+1(2x−3) = (3x+1)(2x−3).
Find and fix the sign error
X2 · WRITTEN EXIT CHECK
A student writes:
6x² − x − 2
= 3x(2x + 1) − 2(2x − 1)
Correct the grouping line. Then factor.
Explain why the sign must change.
Use −2(2x+1), since −2(2x+1)=−4x−2. Result: (3x−2)(2x+1).
Before Wednesday’s quiz
NEXT STEP · END OF CORE LESSON
ALEKS homework R.1–R.5 + Quiz 1
Wednesday, September 2 · 2:10 PM
Practice: grouping → nonmonic quadratics → GCF first.
Then use the full Unit 1A review and checkpoint.
Quiz 1 is in class, on paper.
Check Canvas for the master dates and published coverage.
Use the factors to simplify
OPTIONAL · RATIONAL-EXPRESSION BRIDGE
6x² + 11x + 32x² + 5x + 3
Factor before canceling
Use the same splitting
and grouping method.
Keep every original
denominator restriction.
(3x+1)/(x+1), with x≠−3/2,−1.
Then build an LCD from factors
OPTIONAL · LCD FOLLOW-UP
Denominators:
2x² + 5x + 3 and 2x² + x − 3
Factor each one.
Which factor is shared?
Use each factor to the highest power needed.
LCD = (2x+3)(x+1)(x−1). Original exclusions: −3/2, −1, 1.
Rewrite before adding
OPTIONAL · RATIONAL-EXPRESSION FOLLOW-UP
12x² + 5x + 3
+
12x² + x − 3
Use the LCD from the previous slide.
Rewrite both numerators before adding.
State the original restrictions.
2x/[(2x+3)(x+1)(x−1)], with x≠−3/2,−1,1.
One more factoring challenge
OPTIONAL · EXTRA PRACTICE
12x² − 2x − 4
GCF first.
Find the pair for the remaining quadratic.
Split, group, factor, and check.
2[6x²+3x−4x−2] = 2[3x(2x+1)−2(2x+1)] = 2(3x−2)(2x+1).