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College Algebra K/DCTuesday, 16 September 2014

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

• ASSIGNMENT CHANGE FOR TODAY– Sec. R.3: p. 31 (43-48 all, 50-58 even, 77-85 odd)

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.

2-8

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

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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:

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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

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Multiplying Polynomials

• Find the product.

Use FOIL.

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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!!!

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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

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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

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Multiplying Polynomials

• Find the product.

Multiply the parenthesesfirst. Then distribute.

2-16

Multiplying Polynomials

• Find the product:

Product of the sum and difference of two terms.

2-17

Multiplying Polynomials

• Find the product:

Rewrite as the productof two squares.

Use the box method to multiply.

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Multiplying Polynomials

• Find the product:

HINT: Rewrite.

Now use the box method to multiply.

Assignment: Sec. R.3 – p. 31 (43-48 all, 50-58 even, 77-85 odd) Due on Friday, 19 September 2014.Find each sum or difference.

Find each product.

Perform the indicated operation.

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2 2 5 4 7 53 4 34 ) x x x x 3 2 3 2 3 3 4 2 644) m m m m

2 2 2 12 8 6 4 3 24 45) y y y y 2 246 3 8 5 5 3 2 4) p p p p

4 2 3 2 2 6 3 247) 5 4m m m m m m m m

3 3 2 2 8 3 248) 4 3 1x x x x x x

0) 5 35 6 4m m 1 1 2 3

4 252) m m

3 2 2 4 354) b b b 3 2 2 12 3 156) k k k k

58) 3 2r s t r s t

3 2 2 4 77 3 27 ) p p p p p

9) 7 7 27 2m n m n 2 2 3 4 2 2 41) 38 q q q q

483) 6 2 3 8p p p 2 2 4 6 2 385) y y y y

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College Algebra K/DCWednesday, 17 September 2014

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

• ASSIGNMENTS DUE FRIDAY (QUIZ Day)– Sec. R.3: p. 31 (43-48 all, 50-58 even, 77-85 odd)

• Yesterday’s assignment.– Sec. R.3: p. 31 (87-94 all)

• Today’s book assignment.– WS Sec. R.3

• Today’s worksheet.

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

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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

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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).

Assignment

• Sec. R.3: p. 31 (87-94 all)– Due on Friday, 19 September 2014 (QUIZ

day).

• WS Sec. R.3– Due on Friday, 19 September 2014 (QUIZ

day).

• Also due on Friday, 19 September 2014:– Sec. R.3: p. 31 (43-48 all, 50-58 even, 77-85

odd)2-26

Assignment: Sec. R.3 – p. 31 (87-94 all) Due on Friday, 19 September 2014.Write each problem and perform each division.

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7 6 4 2

2

4 14 10 14

287 )

x x x x

x

3 2 2 38 12 208

48 )

r s r s rsrs

8 6 4

6

10 16 4) 89

2

x x x

x

33)

5

390

2x xx

3 26 7 4 21)

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3m m m

m

4 3 2

2

6 9 292)

8 7

3 2

x x x x

x

4 23 6 9 53)

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3x x x

x

4 2

2

4 2 54

19 )

k k k

k