Transcript 04 Tuesday Factoring Trinomials with a Leading Coefficient

Factoring ax  bx  c
2
Lesson 2.7
Page 85
More Diamonds #1
30
2 x  11
11 x  15
2
2x2
 (2 x  5)( x  3)
+ 5x + 6x + 15
x (2x + 5) + 3 (2x + 5)
(2x + 5)(x + 3)
5
6
• Multiply the coefficient
of x2 and the constant.
• The product goes on
top.
• Place the coefficient of
x on bottom.
• Find the right and left.
• Rewrite the trinomial
with four terms.
• Factor by grouping.
Shortcut Way #1
30
2 x  11
11 x  15
2
 (2 x  5)( x  3)
Step #1: Multiply leading coefficient and constant
together and put on top.
Step #2: Put coefficient of x on bottom.
Step #3: Figure out the left and right numbers to
complete the diamond.
Step #4: Write the answer from the diamond as if
there was a leading coefficient of one. (x+5)(x+6)
Step #5: Divide each number in the binomials by
the leading coefficient. Reduce the fractions if
possible. (x+5/2)(x+6/2) = (x+5/2)(x+3)
Step #6: If you are left with a fraction for the
number in the binomial move the denominator of
the fraction to the coefficient of the variable in the
same binomial. (2x+5)(x+3)
5
6
More Diamonds #2
12
 (2 x  1)(3x  2)
6 x  77 x  2
2
6x2
+ 3x + 4x + 2
3x(2x + 1) + 2 (2x + 1)
(2x + 1)(3x + 2)
3
4
• Multiply the coefficient
of x2 and the constant.
• The product goes on
top.
• Place the coefficient of
x on bottom.
• Find the right and left.
• Rewrite the trinomial
with four terms.
• Factor by grouping.
Shortcut Way #2
12
6 x  77 x  2
2
 (2 x  1)(3x  2)
Step #1: Multiply leading coefficient and constant
together and put on top.
Step #2: Put coefficient of x on bottom.
Step #3: Figure out the left and right numbers to
complete the diamond.
Step #4: Write the answer from the diamond as if there
was a leading coefficient of one. (x+3)(x+4)
Step #5: Divide each number in the binomials by the
leading coefficient. Reduce the fractions if possible.
(x+3/6)(x+4/6) = (x+1/2)(x+2/3)
Step #6: If you are left with a fraction for the number in
the binomial move the denominator of the fraction to the
coefficient of the variable in the same binomial.
(2x+1)(3x+2)
3
4
More Diamonds #3
-18
2 x -33 x  9
2
 (2 x  3)( x  3)
2x2 + 3x - 6x - 9
x (2x + 3) - 3 (2x + 3)
(2x + 3)(x - 3)
3
-6
• Multiply the
coefficient of x2 and
the constant.
• The product goes on
top.
• Place the coefficient of
x on bottom.
• Find the right and left.
• Rewrite the trinomial
with four terms.
• Factor by grouping.
Shortcut Way #3
-18
2 x -33 x  9
2
 (2 x  3)( x  3)
Step #1: Multiply leading coefficient and constant
together and put on top.
Step #2: Put coefficient of x on bottom.
Step #3: Figure out the left and right numbers to
complete the diamond.
Step #4: Write the answer from the diamond as if there
was a leading coefficient of one. (x+3)(x–6)
Step #5: Divide each number in the binomials by the
leading coefficient. Reduce the fractions if possible.
(x+3/2)(x–6/2) = (x+3/2)(x–3)
Step #6: If you are left with a fraction for the number in
the binomial move the denominator of the fraction to the
coefficient of the variable in the same binomial.
(2x+3)(x–3)
3
-6
More Diamonds #4
-12
 (2 x  1)(3x  2)
6 x 1 x  2
2
6x2
- 3x + 4x - 2
3x(2x – 1) + 2 (2x – 1)
(2x – 1)(3x + 2)
-3
4
• Multiply the coefficient
of x2 and the constant.
• The product goes on
top.
• Place the coefficient of
x on bottom.
• Find the right and left.
• Rewrite the trinomial
with four terms.
• Factor by grouping.
Shortcut Way #4
-12
6 x 1 x  2
2
 (2 x  1)(3x  2)
Step #1: Multiply leading coefficient and constant
together and put on top.
Step #2: Put coefficient of x on bottom.
Step #3: Figure out the left and right numbers to
complete the diamond.
Step #4: Write the answer from the diamond as if there
was a leading coefficient of one. (x–3)(x+4)
Step #5: Divide each number in the binomials by the
leading coefficient. Reduce the fractions if possible.
(x–3/6)(x+4/6) = (x–1/2)(x+2/3)
Step #6: If you are left with a fraction for the number in
the binomial move the denominator of the fraction to the
coefficient of the variable in the same binomial.
(2x–1)(3x+2)
-3
4
#5 A Systematic Approach to Finding
the Right & Left
•
•
•
•
•
Multiply the
coefficient of x2
and the
constant.
List all of the
factor pairs of
the product.
Find the pair
that add/subtract
to yield the
middle term.
Rewrite the
trinomial with
four terms.
Factor by
grouping.
6 x  7 x  20
2
6 x  8 x  15 x  20
2 x(
)  5(
)
2 x(3x  4)  5(3x  4)
(3 x  4)( 2 x  5)
2
(6)(-20) = -120
1(120)
2(60)
3(40)
4(30)
5(24)
6(20)
8(15)
10(12)
If the product is positive, then add the factors.
If the product is negative, then subtract the factors.
#6 A Systematic Approach to
Finding the Right & Left
•
•
•
•
•
Multiply the
coefficient of x2
and the
constant.
List all of the
factor pairs of
the product.
Find the pair
that add/subtract
to yield the
middle term.
Rewrite the
trinomial with
four terms.
Factor by
grouping.
2 x  13 x  20
2
2 x  8 x  5 x  20
2 x(
)  5(
)
2 x( x  4)  5( x  4)
( x  4)( 2 x  5)
2
(2)(20) = 40
1(40)
2(20)
4(10)
5(8)
If the product is positive, then add the factors.
If the product is negative, then subtract the factors.
#7
Try it your favorite way!
2
4 x  4 x  35
Answer: (2x–7)(2x+5)
Homework Assignment:
Complete # 8 – 14 on notes handout