5.1 Basic Operations Chapter 5: Algebra 2 Chapter 5: Algebra - PDF document

SET 1 Chapter 5 Algebra لابجــر Chapter 5: Algebra 1

ةيساسلؤا تايلمعلا 5.1 Basic Operations Chapter 5: Algebra 2

Chapter 5: Algebra 3

سـسلؤا نـيناوق 5.2 Laws of Indices Chapter 5: Algebra 4

5.3 Brackets and Factorisation لماوعلا ىلا ليلحتلا و ساوقلؤا عفر Chapter 5: Algebra 5

Chapter 5: Algebra 6

لسلست تايلمعلا ذيفنت 5.4 Order of Operations Chapter 5: Algebra 7

Chapter 5: Algebra 8

5.5 Polynomials دوذـحلا تازـيثك  A polynomial is an algebraic expression in which all terms have variables that are raised to whole number powers.  Polynomial terms cannot contain variables which are raised to fractional powers or terms which contains variables in the denominator.  All the following expressions are polynomials:  x  5 2 3 4 x 2 3  3   6 x y 4 x y xy 8  4 5 z 12 2  xy 4 z  While the following expressions are not polynomials: 3 4  2 x 3 x    2 2 3 x y 4 xy x 8 4  x 2 3  4 x 3  The degree of a polynomial is equal to the degree of the term having the highest degree.  For example, the degree of the first term of is 6 which is higher    2 4 3 2 2 x y 3 x y x y 5 than the degree of each of the other three terms and hence the degree of this polynomial is 6. Example 73 : Determine whether each of the following expressions is a polynomial or not: 3 2   2   4    (a) x 11 x 4 (b) zx 8 x 7 xy (c) 2 y y 8 x 2 (d) 16 4 y  2 4 5   2 2 (e) (f) 2 3 x x 2 xy z Solution:   3 2 (a) is a polynomial x 11 x 4   2 4 zx 8 x 7 xy is a polynomial (b)    (c) 2 y y 8 x 2 is not a polynomial (d) 16 is a polynomial 4 (e)   2 2 x 2 xy is not a polynomial z y  2 4 5 (f) is not a polynomial 2 3 x Chapter 5: Algebra 9

Example 73 : Determine the degree of each of the following polynomials:     2 2 2 5 2 2 2 (a) (b) (c) y (d) 8 and (e) xyz 6 x z 7 x 8 z x 3 x 5 x y 6 6 Solution:   2 2 2 is a fourth degree polynomial (a) 6 x z 7 x 8 z   5 2 2 2 is a fifth degree polynomial (b) x 3 x 5 x y 6 y is a first degree polynomial (c) 8 is a zero degree polynomial (d) 6 xyz is a third degree polynomial (e)  x  Example 73 : Find the sum of 2  x  3 2 and . ( 3 7 3 ) ( x 2 8 ) x Solution:            2 3 2 2 3 2 ( 3 x 7 x 3 ) ( x 2 x 8 ) 3 x 7 x 3 x 2 x 8     2 3 x 7 x 5 x     3 2 7 5 x x x       2 2 2 3 Example 40: Find the sum of and ( 11 5 8 15 ) . ( 13 6 3 9 ) x z xy z x xy Solution:                2 3 2 2 2 3 2 2 ( 13 z 6 x 3 xy 9 ) ( 11 x 5 z 8 xy 15 ) 13 z 6 x 3 xy 9 11 x 5 z 8 xy 15      2 3 2 8 z 5 x 3 xy 6 8 xy      3 2 2 3 xy 8 xy 8 z 5 x 6  2  2  2   3  Example 41: Subtract from ( 11 x 5 z 8 xy 15 ) . ( 13 z 6 x 3 xy 9 ) Solution:                2 2 2 3 2 2 2 3 ( 11 x 5 z 8 xy 15 ) ( 13 z 6 x 3 xy 9 ) 11 x 5 z 8 xy 15 13 z 6 x 3 xy 9      2 2 3 17 x 18 z 8 xy 24 3 xy       3 2 2 3 xy 8 xy 18 z 17 x 24 Chapter 5: Algebra 10

2   z z  3 2 Example 42: Multiply by ( 7 2 4 ) . ( 6 4 x ) xy Solution:          3 2 2 3 5 3 3 2 2 2 ( 6 z 4 x )( 7 xy 2 z 4 ) 42 xyz 12 z 24 z 28 x y 8 x z 16 x (  2  x  Example 43: Divide by 1 ) . ( 3 7 4 ) x x Solution: In division of polynomials, long division is used in the same way it is used in the division of numbers. The result of division of polynomials may or may not contain a remainder and as illustrated in this example and the following examples. Using long division: 3  x 4  2   x 1 3 x 7 x 4 2  3 x 3 x  x 4  4  x 4  4 0       2 Thus ( 3 x 7 x 4 ) ( x 1 ) 3 x 4   3  2   Example 44: Determine ( 2 x 1 ) . ( 4 x 4 x 11 x 9 ) Solution: Using long division: 2 x 2 + 3 x  4  3  2   2 1 4 4 11 9 x x x x x  3 2 4 2 x 6 2   x 11 x 9 6 2  x 3 x  x 8  9  x 8  4  13          3 2 2 Thus remainder 13 , ( 4 x 4 x 11 x 9 ) ( 2 x 1 ) 2 x 3 x 4 13 3  2         or 2 ( 4 x 4 x 11 x 9 ) ( 2 x 1 ) 2 x 3 x 4  2 x 1 Chapter 5: Algebra 11

   5 2 3 23 18 x x x Example 45: Find:  x 2 Solution: Using long division: x 4  2 x 3  2 x 2  4 x  31       5 4 3 2 x 2 x 0 x 2 x 0 x 23 x 18 x  5 4 2 x  4  3  2   2 x 2 x 0 x 23 x 18  x  4 3 2 4 x    2 3 2 x 0 x 23 x 18 x  2 2 3 4 x    4 2 x 23 x 18 4 2   x 8 x 31  x 18 31  x 62  80            5 3 4 3 2 Hence ( x 2 x 23 x 18 ) ( x 2 ) x 2 x 2 x 4 x 31 remainder 80 , 80            5 3 4 3 2 or ( x 2 x 23 x 18 ) ( x 2 ) x 2 x 2 x 4 x 31  x 2 5. 6 Rational Expressions ةـيبسنلا ةيزبجلا زـيباعتلا   2  Fractional expressions such as x 3 and x 4 x 6 are called rational expressions since they  8 x have polynomials as both numerator and denominator.  A rational expression is proper if the degree of the numerator is less than the degree of the denominator.  For example, x is a proper rational expression. 2  x 8  If the degree of the numerator is greater than or equal to the degree of the denominator, then the rational expression is improper. 2 3    For example, x x 2 x 1 and are both improper rational expressions.  2  x 1 x 1 Chapter 5: Algebra 12

Example 46: Determine whether each of the following fractional expressions is a rational expression or not. For rational expressions, determine whether they are proper or improper. 3 2  y 3 x 2 x 5 (a) (b) (c) 2    3 2 2 6 5 8 x y x 1 Solution: 2 3 x (a) is a proper rational expression.  3 6 x 5 3 y (b) is an improper rational expression. 2 2  8 y  2 x 5 (c) is not rational expression. 2  x 1 2 3 A B C    Example 47: Simplify the following: (a) (b)      4 2 5 2 3 4 x x x x x Solution:    2 ( 2 5 ) 3 ( 4 ) 2 3 x x   (a)     x 4 2 x 5 ( x 4 )( 2 x 5 )    4 x 10 3 x 12    ( x 4 )( 2 x 5 )  x 22    2 2 x 3 x 20         A ( x 3 )( x 4 ) B ( x 2 )( x 4 ) C ( x 2 )( x 3 ) A B C    (b)       2 3 4 ( 2 )( 3 )( 4 ) x x x x x x 2    2    2   A ( x x 12 ) B ( x 6 x 8 ) C ( x x 6 )     ( x 2 )( x 3 )( x 4 ) 2    2    2   Ax Ax 12 A Bx 6 Bx 8 B Cx Cx 6 C     ( 2 )( 3 )( 4 ) x x x         2 2 2 Ax Bx Cx Ax 6 Bx Cx 12 A 8 B 6 C     ( x 2 )( x 3 )( x 4 )   2        ( A B C ) x ( A 6 B C ) x ( 12 A 8 B 6 C )     ( x 2 )( x 3 )( x 4 ) Chapter 5: Algebra 13

5. 7 Rationalizing Denominators and Numerators ماقملا وأ طسبلا يف روذجلا نم صلختلا  In working with quotients involving radicals, it is often convenient to move the radical expression from the denominator to the numerator, or vice versa.  This process is called rationalizing the denominator or rationalizing the numerator. Example 48 : Rationalize the denominator or numerator and simplify:  4 y x 1 1 (a) (b) (c)    2 y 8 x x 1 Solution :    x 1 x 1 x 1   (a)  2 2 x 1  x 1   2 x 1  y 8 4 y 4 y   (b)    8 8 8 y y y  4 y y 8   8 y   1 1 1 x x   (c)       1 1 1 x x x x x x   x x 1    x ( x 1 )   x x 1    x x 1   x x 1   1     x x 1 Chapter 5: Algebra 14