1

College Algebra K/DCTuesday, 15 September 2015

• OBJECTIVE TSW add, subtract, mulitply, and divide polynomials.

• ASSIGNMENT DUE– Sec. R.3: pp. 28-29 (1-9 all, 11-22 all, 27-37 all)– Wire basket

• TODAY’S ASSIGNMENT– Sec. R.3: pp. 29-30 (39-57 odd, 77-85 odd)– Due tomorrow/Thursday, 16/17 September 2015.

• QUIZ: M-M-M, Absolute Value will be after the lesson.

3-2

PolynomialsR.3Adding Polynomials ▪ Subtracting Polynomials ▪ Multiplying Polynomials ▪ Dividing Polynomials

2-3

Like Terms

• Like Terms

4x 3 and x 3

5b and 8b

10a 2 and 3a 2

5x 4y 5 and 2x 4y 5

• Not Like Terms

4x 3 and x 2

5b and 8ab

10a 2 and 2a 10

5x 5y 4 and 2x 4y 5

To add or subtract polynomials, add or subtract thecoefficients of like terms.

Like terms are terms with the same variables each raised to the same powers.

2-4

Adding and Subtracting Polynomials

• Add or subtract.

2-5

Adding and Subtracting Polynomials

• Add or subtract.

2-6

Adding and Subtracting Polynomials

• Add or subtract.

2-7

Adding and Subtracting Polynomials

• Add or subtract.

8

College Algebra K/DCWednesday, 16 September 2015

• OBJECTIVE TSW add, subtract, mulitply, and divide polynomials.

• YESTERDAY’S ASSIGNMENT– Sec. R.3: pp. 29-30 (39-57 odd, 77-85 odd)– Due by the end of the period today. wire basket

• TODAY’S ASSIGNMENT– SIGN IN TO ACCOUNT!!!– Sec. R.3: p. 30 (87-94 all)– WS Sec. R.3– Both due tomorrow, Friday, 18 September

2015.

2-9

Multiplying Two Binomials (FOIL Method)

• Multiply

4 3 5x y x y 4 5 4 3 5 3x x x y y x y y

2 220 4 15 3x xy xy y

2 220 19 3x xy y

F O I L

2-10

Squaring a Binomial

25 4p 5 4 5 4p p

225 20 20 16p p p 225 40 16p p

(square the first term) – 2(first term)(second term) + (square the second term)

Always positive

Shortcut:

2-11

Squaring a Binomial

22 4 1x 2 4 1 4 1x x

22 16 4 4 1x x x

232 16 2x x

22 16 8 1x x

2-12

Multiplying Polynomials

• Find the product.

Use FOIL.

2-13

Multiplying Polynomials

• Find the product.

Notice that the two parentheses have the same terms; the only difference is that one is an addition and one a subtraction.

This is a special product of two binomials called the difference of two squares.

The two middle terms will always cancel.

Memorizethispattern!!!

2-14

Multiplying Polynomials

• Multiply

Technique #1: Distribution

2 24 3 2 7 5 3 2 7t t t t t

3 2 212 8 28 15 10 35t t t t t

3 212 23 38 35t t t

2-15

Multiplying Polynomials

• Multiply

Technique #2: Box Method

I do not have a preference as to which methodyou use; you may use whichever is easiest foryou.

3t2 –2t 7

4t 12t3 –8t2 28t

–5 –15t2 10t –35

3 212 23 38 35t t t

2-16

Multiplying Polynomials

• Find the product.

Multiply the parenthesesfirst. Then distribute.

2-17

Multiplying Polynomials

• Find the product:

Product of the sum and difference of two terms.

2-18

Multiplying Polynomials

• Find the product:

Rewrite as the productof two squares.

Use the box method to multiply.

2-19

Multiplying Polynomials

• Find the product:

HINT: Rewrite.

Now use the box method to multiply.

Assignments

• Sec. R.3: pp. 29-30 (39-57 odd, 77-85 odd)– Due by the end of class today (black tray).

• Sec. R.3: p. 30 (87-94 all)• WS Sec. R.3

– Both due Friday, 18 September 2015.

Dividing Polynomials5 4 2

Ex: Perform the division: 10 9 8 8

2.

n n n n

n

5 4 210 9 8 8

2 2 2 2

n n n n

n n n n

One term in the denominator Create separate fractions and simplify.

4 395 4 4

2n n n

Dividing Polynomials7 4

2Ex: Perform the division:

1 9 3

3 .

5k k k

k

7 4

2 2 2

15 9 3

3 3 3

k k k

k k k

5 2 15 3k k

k

Dividing Polynomials3 212 11Ex: Di 5vide by 8 .3 2n n n n

2

3 2

3 2

2

2

4 1

3 2 12 11 5 8

12 8

3 5

3 2

3 8

3 2

10

n n

n n n n

n n

n n

n n

n

n

2 104 1

3 2n n

n

More than one term in the denominator Use long division.

Dividing Polynomials

2-24

3 2Ex: Divide by 4 8 4 6 1 .2m m m m

2

3 2

3 2

2

2

12 3

2

2 1 4 8 4 6

4 2

6 4

6 3

6

1 2

13 2

m m

m m m m

m m

m m

m m

m

m

2 1 13 2

2 32 2 1

m mm

Dividing Polynomials

2-25

3 2 2Ex: Divide by .3 2 150 4x x x

2 3 2

3 2

2

2

3 2

4 3 2 150

3 0 12

2 12 150

2 0 8

12 158

0 0

x

x x x

x x x

x

x x

x x

x

x

Placeholders for missing terms

2

12 1583 2

4

xx

x

Stop! The degree of the remainder (1)Is less than the degree of the divisor (2).