Warm up

Polynomials

Objectives1. Add, subtract, multiply, divide and factor

polynomials2. Simplify and solve equations involving

roots, radicals, and rational exponents3. Perform operations with complex

numbers

5.1 MonomialsVocabulary

Monomial – expression with one term Constants – monomials that contain no

variables Coefficient – the numerical factor of a variable

Degree – the sum of the exponents of the variable

Power – expression of the form xn

Scientific notation – a x 10n where 1< a < 10, it is used to express very large and very small

numbers

Rules for exponents

Negative exponents a–n = (1/an) and

(1/a-n) = an

Product of powers – am x an = am+n

Quotient of powers – (am/an) = am-n

Power of a power – (am)n = amxn

5.1 Examples

Simplify

1. (-2a3b)(-5ab4)

2. (s2/s10)

3. (b2)4

4. (-3c2d5)3

5. (-2a/b2)5

6. (x/3)-4

7. (-3a5y/a6yb4)5

5.1 Examples continued

Express each number in scientific notation

1. 4,560,000

2. .000092 Evaluate

3. (5 x 103)(7 x 108)

4. (1.8 x 10-4)(4 x 107)

5.2 warm up

Top of page 229

1. How can polynomials be applied to financial situations?

2. What is meant by “tuition increases at a rate of 4% per year?

3. Will the amount of the tuition increase be the same each year?

5.2 Polynomials

Vocabulary Polynomial – a monomial or sum of

monomials Binomial – two unlike

terms Trinomial – three

unlike terms

5.2 Examples

Determine whether each expression is a polynomial, state the degree.

1. C4 – 4sqrt(c) + 182. -16p5 + (3/4)p2q7

Simplify3. (2a3 + 5a -7) – (a3 – 3a + 2)4. –y(4y2 + 2y – 3)5. (2p + 3)(4p + 1)6. (a2 + 3a – 4)(a + 2)

5.3 warm up

Top of page 233

1. What does the expression (x/2) shown in the figure represent?

2. What happens to the width of the pipe opening as the length of the pipe increases?

5.3 Dividing polynomials

Simplify polynomial divided by a monomial

Synthetic divisionExamples1. (5a2b – 15ab3 + 10a3b4)/(5ab)2. (X2 – 2x – 15)/(x – 5)3. (x3 – 4x2 + 6x – 4)/(x-2)4. (4y4 – 5y2 + 2y + 4)/(2y-1)

5.4 Factoring polynomials

Factoring Techniques Write rules page 239

1. GCF

2. Difference of 2 squares

3. Sum of two cubes

4. Difference of two cubes

5. Perfect square trinomials

6. General trinomials

7. Grouping

5.4 warm up

Factor

1. 10a3b2 + 15a2b -5ab3

2. x3 + 5x2 – 2x – 10

3. 3y2 - 2y – 5

4. 5mp2 – 45m

5. X3y3 + 8

6. 64x6 – y6

7. Simplify (a2 – a – 6)/(a2 + 7a + 10)

5.5 Roots of real numbers

Warm up p. 244 #57 and #58

Examples

1. (+-) √(16x6)

2. - √(q3+5)4

3. 5√(243a10b15)

4. √-4

5. 6√t6

6. 5√(243(x+2)15)

5.6 warm up

Page 248 #60

5.5 Roots of real numbers

Examples

46 3

10 155

156 6 5

1. 16 2. 5

3. 24 4. 4

5. 6. 243 2

x q

a b

t x

5.6 Radical Expressions A radical expression is in simplified form

when the following conditions are met.

1. The index n is as small as possible

2. The radical contains no factors that are the nth powers of an integer or polynomial

3. The radical contains no fractions

4. No radicals appear in the denominator

5.6 Examples

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