Please open your laptops, log in to the MyMathLab course web site, and open Quiz 6.1.

If you have any time left after finishing the quiz problems, CHECK YOUR FACTORING ANSWERS before you submit the quiz.

Section 6.2

Factoring Trinomials, Part 1

Review problem from Section 6.1:

(Factoring out the GCF and factoring 4-term polynomials by grouping)

Factor completely:

20 3

x y

 30 2

x y

 4 2

x y

 6

xy

Answer: 2

xy

(2

x

 3)(5

x

 1)

How would you check this?

(Because you

WOULD

do that, wouldn’t you???)

Section 6.2

Factoring Trinomials, Part 1

Recall by using the FOIL method that (

x

+ 2)(

x

+ 4) =

x

2 +

4

x

+

2

x

+

8

=

x

2 +

6

x

+

8

So to factor

x

2 + 6

x

+

8

into (x + __ ) (x + __ ), note that

6

is the

sum

of the two numbers

4

and

2,

and

8

is the

product

of the two numbers.

So we’ll be looking for 2 numbers whose product is

8

whose sum is

6

.

and

Note: there are fewer possible pairs of numbers for the product than for the sum, so that’s why we start there first.

Example Factor the polynomial x 2 + 13x + 30.

Since our two numbers must have a product of 30 and a sum of 13, the two numbers must both be positive.

Positive factors of 30 Sum of Factors

1, 30 2, 15 3, 10 1+30=31 1+15=17 3+10=13 So Note, there are other factors (like 6*5), but

once we find a pair that works, we do not have to continue searching.

Now check answer by multiplying

x

2 + 13x + 30

=

(x + 3)(x + 10). the two factors to see if you get back to the original trinomial.

Example Factor the polynomial x 2 – 11x + 24.

Since our two numbers must have a product of 24 and a sum of -11, the two numbers must both be negative.

Negative factors of 24 Sum of Factors -1, -24 -2, -12 -3, -8 -25 -14 -11 So

x

2 – 11

x

+ 24 = (

x

– 3)(

x

– 8). Now check it!

Example Factor the polynomial x 2 – 2x – 35.

Since our two numbers must have a product of -35 and a sum of -2, the two numbers will have to have different signs.

Factors of -35

-1, 35

Sum of Factors

34 1, -35 -5, 7 5, -7 -34 2 -2 So

x

2 – 2

x

– 35 = (

x

+ 5)(

x

– 7).

Check it!

Example Factor the polynomial x 2 – 6x + 10.

Since our two numbers must have a product of 10 and a sum of -6, the two numbers will have to both be negative.

Negative factors of 10

-1, -10

Sum of Factors

-11 -2, -5 -7

Now we have a problem

, because we have exhausted all possible choices for the factors, but have not found a pair whose sum is -6.

So

x

2 – 6

x

+10 is

not factorable

and we call it a

prime polynomial.

Example Factor the polynomial x 2 – 11xy + 30y 2

.

We look for two

terms

whose product is 30

x

2

y

2 and whose sum is –11x

y

. The two terms will have to both be negative. Note: each term will contain the variable

y

, for the sum to be –11x

y

.

Negative factors of 30x 2

y

2 Sum of Factors

-

xy

, -30x

y

-2x

y

, -15x

y

-3x

y

, -10x

y

-31x -17x -13x

y y y

-5x

y

, -6x

y

-11x

y

So

x

2 – 11

xy

+ 30

y

2 = (

x

– 5

y

)(

x

– 6

y

).

Example Factor the polynomial 3x 6 + 30x 5 + 72x 4

First we factor out the GCF . (

Always

check for this first!) 3

x

6 + 30

x

5 + 72

x

4 = 3

x

4 (

x

2 + 10

x

+ 24) Then we factor the trinomial.

Positive factors of 24

1, 24 2, 12

Sum of Factors

25 14 So 3

x

6 + 30

x

5 + 72

x

4 = 3

x

4 (

x

+ 4)(

x

+ 6).

REMINDER:

• • • On a

test or quiz

, if you have time left after finishing all the problems, you should always

check your factoring results by multiplying

the factored polynomial to verify that it is equal to the original polynomial.

Many times you can detect computational errors or errors in the signs of your numbers (i.e those pesky “dumb mistakes”…) by checking your results.

Practice doing this on at least a few homework problems before you hit “Check Answer”, just to make sure you really do know how to check your answers when it comes time for the quiz.

REMINDER:

The assignment on today’s material

(HW 6.2)

is due at the start of the next class session.

Lab hours in 203: Mondays through Thursdays 8:00 a.m. to 6:30 p.m.

Please remember to sign in on the Math 110 clipboard by the front door of the lab

You may now

OPEN

your LAPTOPS

and begin working on the homework assignment.

We expect all students to stay in the classroom to work on your homework till the end of the 55 minute class period. If you have already finished the homework assignment for today’s section, you should work ahead on the next one or work on the next practice quiz/test.