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# Algebraic expressions

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Everything Maths Grade 10. Chapter 1: Algebraic expressions.

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### Algebraic expressions

1. 1. 1 Everything Maths www.everythingmaths.co.za 1. Algebraic expressions Grade 10
2. 2. 2 Everything Maths www.everythingmaths.co.za The real number system The real number system consists of rational and irrational numbers. Rational numbers include the integers, whole and natural numbers. ● Integers are { . . . ; -3; -2; -1; 0; 1; 2; 3; . . . } ● Whole numbers are {0; 1; 2; 3; . . . } ● Natural numbers are {1; 2; 3; . . . }
3. 3. 3 Everything Maths www.everythingmaths.co.za Rational numbers The following are rational numbers: ● Fractions with both numerator and denominator as integers ● Integers ● Decimal numbers that terminate ● Decimal numbers that recur (repeat) A rational number is any number that can be written as a b where a and b are integers and b≠0 Irrational numbers Irrational numbers are numbers that cannot be written as a fraction with the numerator and denominator as integers.
4. 4. 4 Everything Maths www.everythingmaths.co.za Rounding off Rounding off a decimal number allows us to approximate a number. For example to round 2,6525272 to three decimal places: ● count three places after the decimal and place a | between the third and fourth numbers ● round up the third digit if the fourth digit is greater than or equal to 5 ● leave the third digit unchanged if the fourth digit is less than 5 ● if the third digit is a 9 and needs to be rounded up, then the 9 becomes a 0 and the second digit is rounded up So for 2,6525272 we place the marker: 2,652|5272 Then we note that the fourth digit is a 5 so we round up: 2,653
5. 5. 5 Everything Maths www.everythingmaths.co.za Estimating surds If n = 2: ● A perfect square is the number obtained when an integer is squared. ● We can use perfect squares to determine between which two integers a square root lies. If n = 3: ● A perfect cube is the number obtained when an integer is cubed. ● We can use perfect cubes to determine between which two integers a cube root lies. If a and b are positive whole numbers, and a<b , then n √a= n √b 2<√7<3 because 2 2 =4 and 3 2 =9 and √4<√7<√9 2< 3 √10<3 because 23 =8 and 33 =27 and 3 √8< 3 √10< 3 √27
6. 6. 6 Everything Maths www.everythingmaths.co.za Terminology for mathematical expressions A monomial is an expression with one term. A binomial is an expression with two terms. A trinomial is an expression with three terms.
7. 7. 7 Everything Maths www.everythingmaths.co.za Products ● The product of a monomial with a binomial is ax(cx + d) = acx2 + adx ● The product of two binomials is (ax + b)(cx + d) = acx2 + x(ad + bc) + bd ● The product of a binomial and a trinomial is: (A + B)(C + D + E) = A(C + D + E) + B(C + D + E) ● The product of two identical binomials is known as the square of the binomial. ● We get the difference of two squares when we multiply (ax + b)(ax − b) = (ax)2 - b2
8. 8. 8 Everything Maths www.everythingmaths.co.za Factorisation ● Factorisation is the opposite process of expanding the brackets. ● Taking out a common factor is the basic factorisation method. ● We often need to use grouping in pairs to factorise polynomials. ● To factorise a quadratic we find the two binomials that were multiplied together to give the quadratic.
9. 9. 9 Everything Maths www.everythingmaths.co.za Difference and sum of two cubes ● The sum of two cubes can be factorised as: x3 + y3 = (x + y)(x2 − xy + y2 ) ● The difference of two cubes can be factorised as: x3 − y3 = (x − y)(x2 + xy + y2 )
10. 10. 10 Everything Maths www.everythingmaths.co.za Simplification of fractions ● We can simplify fractions by incorporating the methods we have learnt to factorise expressions. ● Only factors can be cancelled out in fractions, never terms. ● To add or subtract fractions, the denominators of all the fractions must be the same. a b × c d = ac bd , (b≠0,d≠0) a b + c b = a+c b , (b≠0) a b ÷ c d = a b × d c = ad bc , (b≠0,c≠0,d≠0)
11. 11. 11 Everything Maths www.everythingmaths.co.za For more practice see: www.everythingmaths.co.za