x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent.

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1 Rules of Exponents: If n > 0, m > 0 are positive integers and x, y are any real numbers, then: x m x n = x m+n x m x n = xm n, if m n (x m ) n = x mn (xy) n = x n y n ( x y ) n = xn y n 1

2 Can we make sense of a negative exponent? E.g. x 5 x 7 = x x x x x x x x x x x x = 1 x 2 If this rule xm x n = xm n is to be true for m < n, then we must have x 5 x 7 = x5 7 = x 2 This implies that we should define x 2 = 1 x 2 Definition of Negative Exponents: If n is any integer and x 0 is a real number, then x n = 1 x n In other words, taking a negative expoenent is the same is taking the reciprocal of the positive expoenent. 2

3 What about x 0? If we want the rule to still hold, we have x 4 x 4 = x4 4 = x 0 but we know that x4 = 1, this naturally leads to: x4 Definition of zero exponent: For all real number x, if x 0, then x 0 = is undefined. 3

4 Simplify Expressions involving integer exponents: In simplifying an expression involving exponents, remember that the Order of Operation still holds and, in the absense of parenthesis, exponents have the highest order of operation. This means that, in the absense of parenthesis, an exponenet is applied only to the number/variable immediately below it. E.g 4 2 = = = = 1 16 ( 4) 2 = 1 ( 4) 2 = x 4 = 3 1 x 4 = 3 x 4 (3x) 4 = 1 (3x) 4 = x 4 = 1 81x 4 3x 4 = 3 1 x = 3 4 x = 3 4 x 4 ( 3x) 4 1 = ( 3x) = 1 4 ( 3) 4 x = x = = (3 0 ) = (1) = 1 ( 3) 0 = 1 2x 0 = 2(1) = 2 (2x) 0 = 1 xy 3 = x 1 y 3 = x y 3 (xy) 3 = 1 (xy) 3 = 1 x 3 y = = =

5 (5 3) 2 = 2 2 = = 1 4 x y 2 = x 1 y = xy2 2 y 1 2 y = xy2 1 2 y 2 (x y) 2 = 1 (x y) = 1 2 x 2 2xy + y 2 (3 2 ) 3 = 3 6 = (x 2 ) 4 = x 8 5

6 Simplify: x 3 x 5 Ans: x 3 x 5 = x 3+ 5 = x 8 = 1 x 8 Simplify: y 2 y 6 = y 2 ( 6) = y 2+6 = y 4 6

7 Simplify: x 3 y 2 x 2 y 1 One Approach: x 3 y 2 x 2 y 1 = x 3 2 y 2 ( 1) = x 5 y 3 = 1 x 5 y3 = y3 x 5 Another Approach: x 3 y 2 x 2 y 1 = y2 y 1 x 3 x 2 = y3 x 5 Simplify: (x 4 y 2 ) 3 Ans: (x 4 y 2 ) 3 = (x 4 ) 3 (y 2 ) 3 = x 12 y 6 7

8 Simplify: ( x y 3 ) 2 Ans: ( ) x 2 = x2 y 3 (y 3 ) = x2 2 y 6 Simplify: Ans: = 32 = 9 Simplify: a 4 a 3 b 2 8

9 Ans: a 4 a 3 b 2 Simplify: ( 5 6) 2 = a7 b 2 = a7 b 2 Ans: ( ) 5 2 ( 6 2 = = 6 5) 62 5 =

10 Simplify: 4x2 y 3 3xy 2 Ans: 4x2 y 3 3xy = 16 9 x2 y 10 = 16x2 y 10 9 Simplify: x 3 y 2 3 5x 1 y 3 Ans: x 3 y 2 5x 1 y 3 ( ) 4 2 ( ) 4 2 ( ) 4 2 = 3 x2 1 y 3 ( 2) = 3 xy5 = (x) 2 (y 5 ) = x1 y 2 y 3 5x 3 3 = (5x2 ) 3 (y 5 ) = (5)3 (x 2 ) 3 = 125x6 3 (y 5 ) 3 y 15 = y5 5x 2 3 = 5x2 y

11 Simplify: 4y 3 y 10 y 2 Ans: 4y 3 y 10 y 2 = 4y y 2 = 4y 13 y 2 = 4y 13 ( 2) = 4y 11 = = y11 y 11 11

12 Simplify: 4x2 y 4 z 4 3x 2 y 1 z 2 Ans: 4x2 y 4 z 4 3x 2 y 1 z ( ) 4 2 = 3 x2 ( 2) y 4 ( 1) z 4 2 ( ) 4 2 ( 4 = 3 x4 y 5 z 6 = 3 x4 y 5 1 ) 2 = z 6 4x4 y 5 3z 6 = 3z6 4x 4 y 5 2 = (3z6 ) 2 2 (4x 4 y 5 ) 2 = (3)2 (z 6 ) 2 (4) 2 (x 4 ) 2 (y 5 ) 2 = 9z12 16x 8 y 10 12

13 Simplify: 3(x 2 ) 3 8(x 4 ) 5 Ans: 3(x 2 ) 3 8(x 4 ) = 3x6 8x 20 = (8x14 ) 2 = (8)2 (x 14 ) 2 (3) = = 64x28 9 ( ) 3 2 = 8x 14 8x

14 Simplify: (a 3 b 4 ) 2 (2a 4 b 4 ) 2 Ans: (a 2 b 6 ) 3 One Approach: (a 3 b 4 ) 2 (2a 4 b 4 ) 2 = (a3 ) 2 (b 4 ) 2 (2) 2 (a 4 ) 2 (b 4 ) 2 (a 2 b 6 ) 3 (a 2 ) 3 (b 6 ) 3 = a6 b a 8 b 8 a 6 b 18 = 2 2 a 6+ 8 b 8+ 8 a 6 b 18 = 2 2 a 2 b 16 a 6 b 18 = 2 2 a 2 ( 6) b 16 ( 18) = 2 2 a 4 b 2 = a4 b 2 = 1 4 a4 b 2 = a4 b 2 Another Approach: (a 3 b 4 ) 2 (2a 4 b 4 ) 2 = (a3 b 4 ) 2 (a 2 b 6 ) 3 = (a3 ) 2 (b 4 ) 2 (a 2 ) 3 (b 6 ) 3 (a 2 b 6 ) 3 (2a 4 b 4 ) 2 (2) 2 (a 4 ) 2 (b 4 ) 2 = a6 b 8 a 6 b 18 4a 8 b 8 = a12 b 10 4a 8 b 8 = 1 4 a12 8 b 10 8 = 1 4 a4 b 2 = a4 b

15 A polynomial function of degree n (in one variable, with real coefficients) is a function of the form: p(x) = a n x n + a n 1 x n 1 + a n 2 x n a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are real numbers. E.g. p(x) = 3x 4 2x is a polynomial of degree 4. f(x) = x x 5 2x 3 + x 5 is a polynomial of degree 10. q(x) = 2x is a polynomial of degree 1. r(x) = 2 is a polynomial of degree 0 (a constant function). 15

16 Notice that a linear function is a polynomial of degree 1. The degree of a polynomial is the highest power of x whose coefficient is not 0. By convention, a polynomial is always written in decreasing powers of x. The coefficient of the highest power of x in a polynomial is the leading coefficient. In the above example, the leading coefficient of p is 3. The leading coefficient of f is 1. The coefficient of a polynomial is understood to be 0 if the term is not shown. A polynomial of one term is a monomial. A polynomial of two terms is a binomial. A polynomial of three terms is a trinomial. A polynomial of degree 1 is a linear function. A polynomial of degree 2 is a quadratic function. A polynomial of degree 3 is a cubic function. A polynomial of degree 4 is a quartic function. A polynomial of degree 5 is a quintic function. 16

17 To add two polynomials, add their terms by collect like terms. E.g. (x 3 +2x 2 3x+4)+( 4x 4 +2x 3 4x 2 +x+1) = 4x 4 +3x 3 2x 2 2x+5 To subtract two polynomials, take the negatives of the second expression and add. E.g. (2x 4 2x 3 + 3x 3) (5x 4 3x 3 + x 2 7x + 6) = 2x 4 2x 3 + 3x 3 5x 4 + 3x 3 x 2 + 7x 6 = 3x 4 + x 3 x x 9 To multiply two polynomials, treat the first polynomial as if it is one single term, and apply the distributive property to distribute the first polynomial to each of the terms of the second polynomial, then multiply each term of the second polynomial to each of the terms in the first polynomial by the distributive property again. 17

18 E.g. (3x 2 + 2x 1)(2x 3) = (3x 2 + 2x 1)(2x) (3x 2 + 2x 1)(3) treat (3x 2 + 2x 1) as a single term and distribute this to (2x 3) by applying the distributive property. 6x 3 + 4x 2 2x 9x 2 6x + 3 multiply 2x to each of the terms of the first polynomial, and multiply 3 to each of the terms in the first polynomial using the distributive property again. Remember to distribute the negative sign as well. 6x 3 5x 2 8x + 3 collect like terms to simplify. There is another method of multiplying polynomials, which is to multiply each term of the first polynomial with each term of the second polynomial: (3x 2 +2x 1)(2x 3) = 6x 3 9x 2 +4x 2 6x 2x+3 (each term of the first polynomial is multiplied to each terms of the second polynomial. Notice that the negative sign is also multiplied. = 6x 3 5x 2 8x

19 In algebra we often need to multiply two binomials (usually of degree 1), using the method of multiplying each term of the first polynomial to each term of the second polynomial we have: (x + 2)( x 4) = x 2 4x 2x 8 The terms that are being multiplied together can be viewed as: The First terms ((x)( x)) because they are the first terms of the two polynomials. The Outer terms ((x)( 4)) because they are the terms on the outside when the expression is written horizontally. The Inner terms ((2)( x)) because they are the terms on the inside when the expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. 19

20 Some special products: (a + b)(a b) = a 2 ab + ab b 2 = a 2 b 2 (difference of two squares) (a + b) 2 = (a + b)(a + b) = a 2 + ab + ab + b 2 = a 2 + 2ab + b 2 (square of a sum) (a b) 2 = (a b)(a b) = a 2 ab ab + b 2 = a 2 2ab + b 2 (square of a sum) 20

21 Factoring Factoring the greatest common factor: E.g. (x 2 4x) = x(x 4) This is the reverse process of the distributive property. x is the GCF of the two terms, and it can be factored out. ( 12x 4 + 6x 2 + 4x) = 2x( 6x 3 + 3x + 2) (4x + 2) = 2(2x + 1) Notice in this example that, if one of the term in an expression is the GCF, when that term is factored out, what remains behind is the number 1. Factoring out the negative sign E.g. (3x 4) = 1( 3x + 4) = (4 3x) ( 2x 3 + 4x 2 2x) = 2x(x 2 2x + 1) 21

22 Factor by Grouping: Factor by Grouping may be applied if an expression to be factored has an even number of terms (4, 6, 8...) and the terms do not share a common factor, but some of the terms have common factors. In this case, group the terms into 2 groups of equal number of terms (2 each if there are 4 terms), where at least one of the groups should have a common factor. E.g. (x 3 +3x 2 +4x+12) = (x 3 +3x 2 )+(4x+12) = x 2 (x+3)+4(x+3) = (x 2 + 4)(x + 3) E.g. ax 4ay + 3bx 12by = (ax + 3bx) (4ay + 12by) notice that we used the method of factoring out the negative sign. = x(a + 3b) 4y(a + 3b) = (x 4y)(a + 3b) 22

23 Factoring trinomial of the form: ax 2 + bx + c If a = 1 We use trial and error method: E.g. x 2 + 5x 6 We are looking for two numbers that multiply to 6 and adds up to 5, Since (6)( 1) = 6 and = 5, 6 and 1 are our choices, we have: x 2 + 5x 6 = (x + 6)(x 1) x 2 + 7x + 12 Two numbers that multiply to 12 and adds up to 7, so 3 and 4 works, we have: x 2 + 7x + 12 = (x + 3)(x + 4) 23

24 If a 1: 6x x 8 The trial and error method still works, but need to be applied more carefully. You choose a factoring of 6, say 6 and 1, to test if it works: 6x x 8 = (6x + a)(x + b) You are looking for a and b such that ab = 8, and 6b + a = 13 (6b + a contributes to the middle term). There is no two numbers a, b that would work. But we are not done, we may try 3 and 2 since (3)(2) = 6. 6x x 8 = (3x + a)(2x + b) We are now looking for a and b such that ab = 8 and 3b + 2a = 13. Notice that a = 8 and b = 1 works: 6x x 8 = (3x + 8)(2x 1) 24

25 AC-method A more systematic method is to multiply a with c, then look for two numbers that would multiply to ac and add up to b, then split the middle term bx into the sum of that two numbers, then factor by grouping: E.g. 12x 2 + x 6 In this example, a = 12 and c = 6. So ac = (12)( 6) = 72. We are looking for two numbers that would multiply to 72 and adds up to the middle term (b = 1). In this example, 9 and 8 work, so we split the middle term like this: 12x 2 + x 6 = 12x 2 + 9x + 8x 6 Then factor by grouping: 12x 2 +9x+ 8x 6 = (12x 2 +9x) (8x+6) = 3x(4x+3) 2(4x+3) = (3x 2)(4x + 3) The sum or difference of two cubes can be factored: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 25

26 Solving Equations by Factoring: The Zero Factor Property of Real Numbers: If ab = 0, then a = 0 or b = 0 (or both = 0). To solve the equation: x 2 + 4x + 3 = 0, we cannot try to isolate the variable by doing just algebraic operations. Instead, we factor the left hand side: (x + 3)(x + 1) = 0 Since (x + 3) and (x + 1) multiplies to 0, the zero factor property tells us that one of them must be 0, we have: x + 3 = 0 or x + 1 = 0 Solving the two equations we get: x = 3 or x = 1 In trying to solve an equation by factoring, one of the sides of the equation (usually the right hand side) must be equal to 0. 26

27 E.g. x 2 6x + 8 = 15 If you try to solve this equation by factoring, you must first move the 15 to the other side to make the right hand side 0. (If you try to factor the expression on the left immediately, you would be wrong). x 2 6x 7 = 0 (x 7)(x + 1) = 0 x 7 = 0 or x + 1 = 0 x = 7 or x = 1 27

28 E.g 4x 2 12x 7 = 0 (2x 7)(2x + 1) = 0 2x 7 = 0 or 2x + 1 = 0 x = 7 2 or x =

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

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