Perform the following operations. 1) (2x + 3) + (4x 5) 2) 2(x + 3) 3) 2x (x 4) 4) (2x + 3)(3x 5) 5) (x 4)(x 2 3x + 5)

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1 2/24 week Add subtract polynomials 13.1 Multiplying Polynomials 13.2 Radicals 13.6 Completing the square 13.7 Real numbers 15.1 and 15.2 Complex numbers 15.3 and 15.4

2 Perform the following operations 1) (2x + 3) + (4x 5) 2) 2(x + 3) 3) 2x (x 4) 4) (2x + 3)(3x 5) 5) (x 4)(x 2 3x + 5)

3 Vocabulary Polynomial Term Constant Monomial Coefficient Binomial Trinomial

4 (x 3) can be thought of (x + (-3)). (x + 3) can be thought of and written as (-1)(x + 3) (x + 3) can also be thought of as take the opposite sign of everything inside the parentheses So: 5 (3x 4) = 5 + (-1)(3x 4) = 5 + (-3x + 4) = x + 4 = 5 3x + 4

5 Add / Subtract Rules + Add + = add numbers and keep + - add -, add numbers and keep + Add - => Subtract numbers and keep sign of the largest number Add Big +? -? Big -

6 Multiplication Rules + times + = + - times - = + + times - = - - times + = - Times

7 Day 2: Number System Make Venn diagram for numbers Imaginary Numbers: i is Modulo 4 simplify 1 Complex numbers are of the form a + bi, where a is the real part and b is the imaginary part Add and subtract complex numbers i n i

8 Day 2: Number System Multiply complex numbers Distribute as with a variable Remember, i 2 = -1 Do examples, including conjugates a + bi and a bi are called conjugates of each other

9 Day 2: Number System Complex number plane a + bi = distance, found by using Pythagorean theorem

10 Day 3 Warm Up Multiply the following: (2 + 3i)(2 3i) (-4 5i)(-4 + 5i)

11 Day 3 1) Conjugates MCC9-12.N.CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 2) Number sets are closed or not closed MCC9-12.A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. (Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.)

12 Day 3: Number System Divide complex numbers Divide normally if imaginary part is 0, (b = 0) Cannot leave i in the denominator since it is a square root. Eliminate i in the denominator by: If only imaginary, (a = 0), then multiply numerator and denominator by I If real and imaginary in denominator, multiply numerator and denominator by the conjugate of the denominator (a + bi and a bi are conjugates)

13 Property of Closure A number set is closed under an operation if the operation is completed on two numbers taken from the set, and it generates another number in the set. Real numbers are closed under multiplication. Are real numbers closed under addition and subtraction? Yes Are real numbers closed under division? No, because you cannot divide by zero Are real numbers closed under the square root operation? No, because the square root of -1 is not a real number.

14 WATCH FOR THE EXCEPTIONS, WHICH ARE OFTEN TIMES ZERO OR -1 HW: Skills page 977 # 8, 9 & 11

15 Day 4 Warm Up NEED TO ADD OPERATIONS OF RATIONAL AND IRRATIONAL NUMBERS WHEN ARE THEY IRRATIONAL OR RATIONAL? Multiply (a + b)(a b) What is the short cut? What does this process do to b? What if b is a square root or i?

16 Justify Operations MCC9-12.N.CN.2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

17 Identity from When you use an operation to combine an identity with another number, that number stays the same. Zero is the additive identity: 0 + a = a = a is the multiplicative identity: 1 * a = a = a * 1.

18 Associative Property Associative Property has to do with grouping Think when you associate with friends. You have a number of friends at school and church. You can group you and your church friends and then the others, or you can group you, your school friends and then the others. It is still your same group of friends Associative Property of addition: a + (b + c) = (a + b) + c What would be the associative property of multiplication? Is subtraction associative? No: 2 (3 5) (2 3 ) - 5 How about division? No: (12/6)/2 12/(6/2)

19 Commutative Property Commutative Property has to do with order Think when you commute from home to school and visa versa they are the equal Commutative Property of addition: a + b = b + a What would be the commutative property of multiplication? Is subtraction commutative? Is division commutative? No No

20 Distributive Property Distributive Property of Multiplication over Addition and Subtraction: For all real numbers a, b, and c: a(b + c) = ab + ac a(b c) = ab - ac Note the order!! Most mathematicians insist on this order.

21 Inverse Property The inverse of an operation undoes the operation: division undoes multiplication. A number's additive inverse is another number that you can add to the original number to get the additive identity. For example, the additive inverse of 67 is -67, because = 0, the additive identity. What would be a multiplicative inverse? Zero does not have a multiplicative inverse, since no matter what you multiply it by, the answer is always 0, not 1.

22 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement (a + b) + (-a) = (b + a) + (-a) Reason

23 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add

24 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)]

25 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add

26 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0

27 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0 Additive Inverse Prop

28 Example of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0 Additive Inverse Prop = b

29 Examples of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0 Additive Inverse Prop = b Identity Prop of Add

30 Examples of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0 Additive Inverse Prop = b Identity Prop of Add (a + b) + (-a) = b

31 Examples of Justifying Steps Prove: (a + b) + (-a) = b, justify each step Statement Reason (a + b) + (-a) = (b + a) + (-a) Commutative Prop of Add = b + [a + (-a)] Associative Prop of Add = b + 0 Additive Inverse Prop = b Identity Prop of Add (a + b) + (-a) = b Transitive Prop of Eq.

32 Do problems in Vol 2 page 1088 Do problems in Student Assignments page 254 Hw Skills book page 975 all, 981 all and page # 13, 19, 21

33 Day 5 Warm Up Turn in signed grade letter Simplify:

34 Irrational Number Operations EOCT Study Guide page 107, questions page 110 The sum, product, or difference of two rational numbers is always a rational number. The quotient of two rational numbers is always rational when the divisor is not zero. The product or the sum of a rational number and an irrational number is always irrational It depends on the numbers if the product of two irrational numbers is rational or irrational.

35 Day 5: Exponent Rules 1. a 1 = a, a 0 2. Zero Exponent a 0 = 1, a 0 m 3. Negative Exponent a, a 0 1 a m

36 Day 5: Exponent Rules 4. Product of Powers: a m * a n = a m + n 5. Power of a Power: 6. Power of a Product m mn a n a m m m ab a * b

37 Day 5: Exponent Rules 7. Quotient of Powers: a a m n a ( m n) 8. Power of a Quotient: a b m a b m m, b 0

38 Day 5: Exponent Rules Index Base Exponent Index Base Exponent Base Exponent Index b x a x a b b x a

39 Simplify radicals, including higher indexes than 2 Rational exponents are a way to express roots as powers. Rationalize denominator with radicals, then imaginary numbers Add and subtract like radicals using distributive property, like sqrt 8 + sqrt 2 Rewrite expression using only positive rational exponents

40 HW: Handout on simplifying rational exponents

41 EOCT Study Guide I see a LOT of proving numbers are rational or not I see some rationalizing denominator with square roots of natural numbers, but I do NOT see any rationalizing the denominator with i Looks like we need to justify each step, associative, commutative, etc.

42 EOCT Released Questions Need equation of the circle Need union and intersection nomenclature and

43 Changing from Decimal to Fraction Rational numbers either terminate or repeat. To change a rational number that terminates to a fraction, just write it as you say it Example: Convert the following to a fraction: = 1/2 = 1/8

44 Changing from Decimal to Fraction Rational numbers either terminate or repeat. To change a rational number that terminates to a fraction, just write it as you say it To change a repeating decimal to a fraction: Let the number = x Determine by how many positions the decimal takes to repeat. Multiply x by the number it takes to repeat - y Subtract y - x Solve for x

45 = 1/13 HW: Skills page 979 # all

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