Transcript Factoring Polynomials

Bell Ringer
1. What is a factor
tree?
2. What are the
terms at the bottom of
a factor tree called?
3. What is GCF?
Factoring Polynomials
Thursday October 2, 2014
Greatest Common Factor
No matter what
type of polynomial
you are factoring,
you always factor
out the GCF first!
What if it’s a binomial?
1st – Factor out GCF
2nd – Difference of Squares
3rd – Sum of Cubes
4th – Difference of Cubes
Binomials continued …
Difference of squares –
Ex: (4x2 – 9)  (2x + 3) (2x – 3)
Sum of cubes –
Ex: 8x3 + 27  (2x +3) (4x2 – 6x + 9)
Difference of cubes –
Ex: x3 – 8  (x – 2) (x2 + 2x + 4)
What if it’s a trinomial?
1st – Factor out GCF
2nd – Perfect Square Trinomial
3rd – “Unfoil”
Trinomials continued…
 1st term is a perfect square, last term is a perfect square,
middle term is double the product of the square roots of the
first and last terms. Then, subtract or add depending on sign
of middle term.
 Ex: 4x2 – 4x +1  (2x -1)2
Square root of 4x2 is 2x, square root of 1 is 1, 2(2x * 1) = 4x
 Ex: 9x2 + 24x + 16  (3x + 4)2
Square root of 9x2 is 3x, square root of 16 is 4, 2(3x * 4) = 24x
Trinomials continued… “Unfoil”
 Find the factors of the first and last terms.
How can we get the middle term with
them?
 If it’s a + and + or a – and +, you need to
multiply and then add to get the middle
term. You will factor as a - - or a + +.
 If it’s a + and -, then you need to multiply
then subtract to get the middle term. You
will factor as a + -.
Examples:
If it’s a + and + or a – and +, you
need to multiply and then add to get
the middle term. You will factor as a
+ + or a - -.
a2 + 7a + 6 = (a + 6) (a + 1)
x2 – 5x + 6 = (x – 3) (x – 2)
Examples:
If it’s a + and -, then you need
to multiply then subtract to get
the middle term. You will factor
as a + -.
x2 + 4x – 5 = (x + 5) (x – 1)
Uncover the mystery
of factoring complex
trinomials!
Tic-Tac-But No Toe
Part 1: In the following tic tac’s there are four numbers. Find the
relationship that the two numbers on the right have with the two
numbers on the left.
-90
10
36 -6
-36 -6
-30 -6
1
-9
-12 -6
0
6
-1
5
-49
7
120
30
-81
9
-24
-6
0
-7
34
4
0
-9
-10 -4
-72
24
16
4
-6 -3
49
-7
21
-3
8
4
-1
-14
-7
1.
2.
2
What did you find?
Did it follow the pattern every time?
Tic-Tac-But No Toe
Part 2: Use your discoveries from Part 1 to complete
the following Tic Tac’s.
9
16
18
6
-35
10
-10
9
7
2
4
45
6
-3
-15
-5
14
-5
-2
2
72
-6
-72
-36
-22
-38
-5
-1
5
9
3.
Did your discovery work in every case?
4.
Can you give any explanation for this?
Finally!
Factoring with a Frenzy!
 Arrange the expression in descending (or
ascending) order.
ax2 + bx + c = 0
 Be sure the leading coefficient is positive.
 Factor out the GCF, if necessary.
 Multiply the coefficients “a” and “c” and put
the result in quadrant II of the Tic Tac.
 Put the coefficient “b” in quadrant III of the
Tic Tac.
 Play the game! Just like the previous
problems. (Find the relationship!)
Once you have completed
your Tic Tac–
WHERE’S the ANSWER?
Use the “a” coefficient as the numerator of
two fractions. Use the results in quadrants I
and IV as the two denominators.
Reduce the fractions.
The numerator is your coefficient for x in your
binominal and the denominator is the
constant term.
EXAMPLE: If you get the fractions ½ and
-3/5, your answer would be (x + 2) (3x – 5).
EXAMPLES
X2 – X - 12
-12 ?
-1
-12 3
-1
-4
?
What 2 numbers
complete the Tic Tac?
Since a = 1, put a 1 in for the
numerator in two fractions.
You found 3 and -4. These are the
denominators for the two fractions.
Your fractions are 1/3 and –1/4
Your answer is (x + 3) (x – 4).
EXAMPLES
2X2 + 8X - 64
*Remember that
-32 ?
What 2 numbers
sometimes a GCF
complete the Tic Tac?
should be factored
4
?
out before beginning.
2(X2 + 4X – 32)
Since a = 1, put a 1 in for the
numerator in two fractions.
-32 8
4
-4
You found 8 and -4. These are the
denominators for the two fractions.
Your fractions are 1/8 and –1/4.
Your answer is 2 (x + 8) (x – 4).
EXAMPLES
1/2X2 + 1/2X - 6
*Remember that
-12 ?
What 2 numbers
sometimes a GCF
complete the Tic Tac?
should be factored
1
?
out before beginning.
1/2(X2 + X – 12)
Since a = 1, put a 1 in for the
-12 -3 numerator in two fractions.
1
4
You found -3 and 4. These are the
denominators for the two fractions.
Your fractions are –1/3 and 1/4.
Your answer is ½ (x – 3) (x + 4).