PreCalc 11

4.1 Factoring Polynomial Expressions (Day 1)

I. Factoring a difference of squares: the form !"#" − %"

Remember the conjugate equations:

We can use these to find the factors of polynomials that are a difference of squares:

i.e. &' − 100

Example:

1. 4,' − 25

Step 1: Look for a binomial (2 numbers) with a minus sign between the two

Numbers.

Step 2. Check if all numbers in the equation perfect squares?

Step 3: Write two sets of brackets. In the first set of brackets write the difference of the roots of both terms

Step 4: Write the conjugate of the first set of brackets in the second set of

Brackets.

Step 5: Check answer by expanding the brackets

i

z g

4h2 lbzeta se a

2e q2

x 10 21 10

check 2n 5 Ints2h 5 Ints 4h2 Yon ion 25

4h2 25

PreCalc 11

2. 160' − 81

On your Own:

3. 36&' − 49

II. Trinomials with leading coefficient = 1 i.e. the form: #" + %# + 5

Recall:

Example:

1. &' + 6& + 9

Step 1: List out all of the factors of c

Step 2 : Pick two factors that add to b and multiply to c

Step 3 : Write two sets of brackets multiplied together. In each set of brackets write the sum of x and one of the factors of c that you selected in step 2.

Step 4 : Double check your answer by expanding the equation

Check 4g 9 4gt94y 9 49 9

165 365316 81

165 81

check Gu 7 GattGr 7 62 7 3622 422 422 49

365 49

3 terms

B at 3 Check Get3 sets11330 221321 32 9I 9 settlor 9

PreCalc 11

2. &' + 4& − 21

On your Own

3. &' + & − 12

III. Trinomials with leading coefficient not = 1 i.e. the form: !#" + %# + 5

Step 1: Make sure that the trinomial is written in the correct order; the trinomial must be written in descending order from highest power to lowest power.

Step 2 : Decide if the three terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 3 : Multiply the leading coefficient and the constant, that is multiply the first and last numbers together.

Step 4 : List all of the factors from Step 3 and decide which combination of numbers will combine to get the number next to x.

Step 5 : After choosing the correct pair of numbers, you must give each number a sign so that when they are combined they will equal the number next to x and also multiply to equal the number found in Step 3.

Step 6 : Rewrite the original problem with four terms by splitting the middle term into the two numbers chosen in step 5.

Step 7 : Now that the problem is written with four terms, you can factor by grouping.

Check7 K 3 Get7 K 3

21 I7 3 22 321 72 21

22 the 21

Cheek

21 4 2 3 2 4 2 3

12 NZ 321442 iz2 63 4 22 th 12

2x the 621272k 3

a c

PreCalc 11 Example:

1. 2&' − 9& − 5

2. 6&' + & − 2

On your Own

3. 3&' − 2& − 8

Homework: Pg 181 # 3-10

2 9z gCheck

z L Sr 22 1 z s

2K 10K 1 be 5 2n lortz S

2 5 10 2n K s t l a s 22 ga of11

i lo2 E 2x 11 n s

622 2 2Check

d d 12K 1 32126221424 32 2 622142 32 2

x 2 12 2213k t2 1 32 2 Galta 21 122 6 2K 1 321 23 4

3 2x 8 Check

32.2 Gk se g3214 se z

3 s 24 342 2 14 2 2322 62 42 g

l 3kt 4 se 2322 2x 8

1 242 123846

Factoring W s