Warm up

Polynomials Objectives Add, subtract, multiply, divide and factor polynomials Simplify and solve equations involving roots, radicals, and rational exponents Perform operations with complex numbers

5.1 Monomials Vocabulary 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 How can polynomials be applied to financial situations? What is meant by “tuition increases at a rate of 4% per year? 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) + 18 2. -16p5 + (3/4)p2q7 Simplify 3. (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 What does the expression (x/2) shown in the figure represent? 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 division Examples (5a2b – 15ab3 + 10a3b4)/(5ab) (X2 – 2x – 15)/(x – 5) (x3 – 4x2 + 6x – 4)/(x-2) (4y4 – 5y2 + 2y + 4)/(2y-1)

5.4 Factoring polynomials Factoring Techniques Write rules page 239 GCF Difference of 2 squares Sum of two cubes Difference of two cubes Perfect square trinomials General trinomials 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 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

5.6 Radical Expressions A radical expression is in simplified form when the following conditions are met. The index n is as small as possible The radical contains no factors that are the nth powers of an integer or polynomial The radical contains no fractions No radicals appear in the denominator

5.6 Examples Overhead projector