5.1 Monomials. Algebra 2

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1 . Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific Notation Example : Write each number in scientific notation.,00,000 B. 7,00, D Example : Write each number in standard form..0 0 B D Rules for Multiplying Monomials:.. Example : Simplify a b a b c B. x yz x y z a b 6a b c a bc 7 Rules for Dividing Monomials:... Example : Simplify x x B. x y xy 7 xy xy 8 D. 7a b c d a b c

2 Properties of Powers:... Example : Simplify each expression. a B. b c x y z 6x yz D. f g f g E. 7 n m p q F. 8x y b xy b c c G. x y z 6x yz 0 Example 6: Evaluate. Express each answer in both scientific and decimal notations B ,000, D., 000,

3 . Polynomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Polynomials A polynomial is a or. x x x and x x and 7 x x y x y x are polynomials. are NOT polynomials Degrees The degree of a term is the. The degree of a polynomial is the of the monomial with the. x x x is a polynomial of degree. x y x y x is a polynomial of degree. Example : Determine whether or not each expression is a polynomial. Then state the degree of each polynomial. 7 x x x B. x y x y x y x x D. x x x Like Terms: Like terms are terms that have. Example : Simplify. 7xyz xyz xyz B. x x 0x 8x x x x x xy y x xy y D. 9a a 7b E. x 7x x x

4 FOIL Method: F: First: O: Outer: I: Inner: L: Last: Example : Find each product. x 8 x B. a a b b 6 D. x 7 E. x F. x x x Example : Find the perimeter of the figure. ( 6a - ) ft a ft ( a + ) ft

5 . Dividing Polynomials Algebra Goal : A..: Divide a Polynomial by a monomial. Dividing a Polynomial by a Monomial:. Split the polynomial into parts at the addition and subtraction signs.. Write each part over the denominator.. Divide (reduce) each part by the denominator. Example : Simplify. x y x x B. 6 8 a b a b ab ab 6x y 8x y 80x y 8xy Synthetic Division:. Write the polynomial in descending order.. Write in the form shown below. Write the coefficients of each term and write in zeros for any missing terms. The Zero Polynomial written without variables in Descending order. Quotient...(polynomial answer) Remainder. Write the first coefficient below the line.. Multiply the number below the line by the zero in the box at the top left. Write the answer under the second coefficient above the line.. Add the numbers in the second column and write the answer below the line. 6. Repeat steps and until you get the remainder. 7. Your answer is below the line it is a polynomial with a degree on less than the original polynomial. 8. If the remainder is not zero, write the remainder as the numerator and the divisor as the denominator. Example : Use synthetic division to find x x x x.

6 c Example : Use synthetic division to find c c 0 6. Example : Use synthetic division to find x x x x x x. Example : Use synthetic division to find h h 8 h. Example 6: Use synthetic division to find 6n n n. Example 7: Use synthetic division to find x x x 7 x.

7 Long Division: Example 8: Use long division to find x x x. Example 9: Use long division to find x x x x Example 0: Use long division to find x x. x Example : Use long division to find 8x 8x 6x 6 x

8 Greatest Common Factor (GCF): Example : Factor.. Factoring Algebra 6m n mn B. k p k p k p a a 8 D. k 60k 7k Grouping: Example : Factor. 7b b b B. a x b x a y b y Difference of Squares: There must be subtraction problem not an addition problem. Example : z 8 B. c 9 9a D. x 8 E. 6z 6 Difference and Sum of Cubes: Be careful with the signs in your answer. Example : x 7 B. x 7 8x

9 . Roots of Real Numbers Algebra Goal : Simplify radicals having various indices. Goal : Use a calculator to estimate roots of numbers. n = Example : Find each root. = 9 = 8 = 7 = 6 = = n b n b > 0 b < 0 b = 0 Example : Find each root. 69x = B. 8 (8 x - ) = a 6 = D. mn =

10 Problems with an that have an answer with an an sign must be used. It identifies the. Example : ( an ) = B. 6 6 ( xy ) = 6 8 ( - y ) = D. 6 x = E. 6 = F. 6 =

11 .6 Radical Expressions Algebra Goal: Simplify radical expressions. Goal: Rationalize the denominator. Goal: Add, subtract, multiply and divide radical expressions. Example : Simplify 0a B. 7 x y z 7 a b Example : Simplify 0 x y 0xy B. m n 6mn t 8t v Rationalizing the denominator:

12 Example : Simplify 0 8 B. 6 kn D. 7x Adding and Subtracting Radical Expressions: Example : Simplify B Example : Simplify ( 6 + )( + ) B. ( + )( + 6 )

13 Conjugates: Example 6: Simplify + - B

14 .7 Radical Expressions Algebra Goal: Write expressions with rational exponents in simplest radical form and vice versa. Goal: Evaluate expressions in either exponential or radical form. Definition of n b : For any real number b and for any integer n >,. except when b < 0 and n is even. Remember negative exponents:. Example : Evaluate. 6 B. 6 9 D. 8 E. 6 F. 7 8 Definition of Rational Exponents: For any nonzero real number b, and any integers m and n, with n >, m n b =. except when b < 0 and n is even. Example : Write each expression in multiple ways. 7 x B. y a

15 Example : Simplify each expression. 7 x x B. a a x x x D. w E. z 7 F. xy 8 G. x y H. a I. 7 Example : Simplify each expression. 6 a b B D. 6 n E. n F. a a a G. a b H. 8 I.

16 .8 Solving Equations Containing Radicals Algebra Goal: Solve equations containing radicals. Example : Solve each equation. x 8 9 B. x 9 0 Example : Solve each equation. y 0 B. t 7 Example : Solve each equation. y 0 B. t

17 Example : Solve each equation. x x B. n 9n 9 Example : Solve each equation. x x B. a a 7

18 .9 Complex Numbers Algebra Goal: Simplify square roots containing negative radicals. Goal: Solve quadratic equations that have pure imaginary solutions. Goal: Add, subtract, and multiply complex numbers. Definition of Pure Imaginary Numbers For any real number b, = = Where i is the unit, and bi is called an. Definition of a Complex Number A complex number is written in the form, where and are and is the. is called the part and is called the part., i Example : Simplify each expression. 8 B. 00x x Example : Simplify each expression. 8i i B. i 7i 0 D. 7 8

19 Example : Solve each equation. x 8 0 B. a 7 0 Example : Simplify each expression. 8 7i i B. 9 6i i Example : Simplify each expression. 8 i i B. 6 i i 6 i i D. 7i

20 .0 Simplifying Expressions with Complex Numbers Algebra Goal: Simplify rational expressions containing complex numbers in the denominator. Complex Conjugate: For a bi the conjugate is. For a bi the conjugate is. For bi the conjugate is. For bi the conjugate is. Example : Find the conjugates of each complex number. i B. i 7i D. 8 i Example : Find the product of each complex number and its conjugate. 8i B. 7i i D. i Example : Simplify. 8 i B. 7 i i i i D. i i E. i F. i

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