ALGEBRAIC EXPRESSIONS Real numbers Surds lie between integers Rounding-off Integers Products Factorization: Common Factor Factorization: Difference of Squares Factorization: Trinomials Factorization: Grouping in Pairs Factorization: Cubes Algebraic Fractions Algebraic Language

REAL NUMBERS Rational Numbers A rational number is a number that can be expressed in the form where and where a and b are integers.

Examples a) Integers e.g. 5 can be written as where 5 and 1 are integers. b) Mixed fractions e.g. c) Terminating decimals e.g. 0,25 = d) Recurring decimals have an infinite pattern & can be expressed as a fraction e.g. 0, 3 = 0, 3333333; 0, 12 = 0, 12121212

Converting Recurring Decimals to Fractions E.g. a) Show that 0,3 is rational. Let x = 0, 333333.... l0x = 3, 333333 (multiply both sides by 10) l0x - x = 3, 33333 – 0, 33333 (subtract equations) 9x = 3, 0000000 9x = 3 x = 3 … a rational number! E.g. b) Show that 0, 12 is rational. Let x = 0, 12121212 100x = 12, 12121212 (multiply both sides by 100) 99x = 12, 000000 (subtract equations) x = … a rational number!

EXERCISE 1. Are these numbers rational and why? (a) (b) (c) 6 (d) - 3 (f) 1, 4142

2. Show that the following recurring decimals are rational: (a) 0, 4 (b) 0, 21 (c) 0, 14 (d) 19, 45 (e) 0, 124 (f) 0, 124

Irrational Numbers in Circles & Squares Numbers that cannot be written in the form where Therefore recurring numbers that neither terminate nor recur with a pattern E.g. a) 5,739129… b) -4,883291103… c) Irrational Numbers in Circles & Squares

The Number is the ratio of the circumference of a circle to its diameter is = 3,142857143…. However, can be approximated as an improper fraction Rounding-off π π as a Rational Number

EXERCISE 1 State whether the following numbers are rational or irrational: (a) 8 (b) (c) 7 (d) (e) (f) (g) (h) (i) - 0, 13 (j) 0, 42 (k) (l) 0, 2453756…

EXERCISE 2 Classify numbers by placing ticks in the appropriate columns: Real Rational Integer Whole Natural Irrational - 3 0, 3 8, 23647

SURDS LIE BETWEEN INTEGERS E.g. Determine without the use of a calculator, between which 2 integers lies. Find an integer smaller and bigger than 11 that can be square rooted … 9 and 16 Now create an inequality … 9 < 11 < 16 Square root all integers … Solve … Check using a calculator …

EXERCISE Without using a calculator, determine between which two integers the following irrational numbers lie: (a) (b) (c) (d) (e) (f)

ROUNDING-OFF INTEGERS If it is > 5 or = 5 … round up If it is < 5 … round down Remember! If you are rounding-off to 2 decimal places, the third decimal place determines whether you round up or down etc. E.g. (a) 2, 31437 (2 d.p.) … Answer: 2, 31 (b) 0, 77777 (3 d.p.) … Answer: 0, 778 (c) 245, 13589 (4 d.p.) … Answer: 245,1359 Rounding-off Numbers

EXERCISE 1 Round off the following numbers to the number of decimal places indicated: (a) 9, 23584 (3 decimal places) (b) 67, 2436 (2 decimal places) (c) 4, 3768534 (4 decimal places) (d) 17,247398 (5 decimal places) (e) 79, 9999 (3 decimal places) (f) 34, 2784682 (4 decimal places) (g) 5,555555 (5 decimal places)

EXERCISE 2 Simplify and round-off to the number of decimal places indicated: (a) (3 decimal places) (b) (4 decimal places) (c) (2 decimal places) (d) (5 decimal places) (e) (2 decimal places)

PRODUCTS E.g. x (y + z) = xy + xz - Multiply each term inside the bracket by the number outside the bracket E.g. (a + b)(c + d) = ac + ad + bc + bd - This is done by using the FOIL method The Product Game

Squaring a Binomial Example Examples Expand and simplify the following: (a) (x + 2) (x + 3) (b) (x + 2) (x² + x - 1) Squaring a Binomial Example

(c) x(x²-2xy+3y²) - 2y(x² -2xy+3y²) (d) (a – 3b) (a – 3b)²

Exercise 1: Simplify: (a) (x + 3)(x - 3) (b) (x - 6)(x + 6) (c) (2x - l)(2x + l) (d) (4x + 9)(4x - 9) (e) (3x - 2y)(3x + 2y) (f) (4a³ b + 3)(4a³ b - 3) (g) (2x – 3 + y)(2x – 3 – y) (h) (1 – a )(1 – a )(1 + a)

Exercise 2 Simplify: (a) 2x(3x - 4y)² - (7x - 2xy) (b) (5y + 1)² - (3y + 4)(2 - 3y) (c) (2x + y) - (3x - 2y) + (x - 4y)(x + 4y) (d) (8m - 3n)(4m + n) - (n - 3m)(n + 3m) (e) (3a + b)(3a - b)(2a + 5b)

Exercise 3 Simplify: (a) (x + 1)(x² + 2x + 3) (b) (x - 1)(x² - 2x + 3) (c) (2x + 4)(x² - 3x + 1) (d) (2x - 4)(x² - 3x + 1) (e) (3x-y)(2x² + 4xy – y² ) (f) (3x - 2y)(9 x² + 6xy + 4 y² ) (g) (3x + 2y)(9x² - 6xy + 4y² ) (h) (2a + 3b)² (i) (2a² - 3b)²

FACTORIZATION: Common Factor The Factor Game The golden rule of factorization is to always look for the highest common factor first:   Basic examples Common Factor with Variables Complex example e.g. a(x-y) – 2(x-y)² = (x-y)[a-2(x-y)] = (x-y)(a-2x+2y)

Common Brackets Exercise: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o)

FACTORIZATION: Difference of Squares There must be 2 terms that you can take the square-root of and a minus sign. Basic examples a) b) Difference of Squares Example

Complex examples a) b) c)

Complex Exercise (a) (b) (c) (d) (e) (f) (g) (n)

FACTORIZATION: Trinomials Make sure you know your times-tables and factors! E.g. a) Factors of 8: 1 x 8 or 4 x 2 The middle term (6a) is obtained by adding the factors of 8 … 4 + 2 = 6 Therefore: E.g. b) First take out common factor! The middle term (7x) is obtained by adding the factors of 8 … -8 +1 = -7 Trinomial with Common Factor

Visualizing Factorization Note: If the sign of the last term of a trinomial is positive, the signs in the brackets are the same i.e. (… - …)(… - …) or (… + …)(… + …) If the sign of the last term of a trinomial is negative, the signs in the brackets are different, i.e. both positive or both negative i.e. (… + …)(… - …) or (… - …)(… + …) Visualizing Factorization

Basic Exercise Factorize fully : (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (o)

Complex Exercise Factorize fully: (a) (b) (c) (d) (e)

More advanced trinomials E.g. a) Step 1: Check for the HCF … none Step 2: Write down the brackets and the factors of the first term and the factors of the last term … (7p 1)(3p 4) Step 3: Now multiply the innermost and the outermost terms … Step 4: To find the middle term ... - 3p + 28p = + 25p Step 5: Complete the factors … (7p – 1)(3p + 4) Note! This method involves trial and error and you need to keep t trying different options until you get ones that will work.

E.g. b) Step 1: Check for the HCF … none Step 2: Write down the brackets and the factors of the first term and the factors of the last term … (12a b)(2a b) Step 3: Now multiply the innermost and the outermost terms … 2ab x 12ab Step 4: To find the middle term ... -12ab + 2ab = - 10ab Step 5: Complete the factors … (12a + b)(2a – b)

Advanced Exercise 1. Factorize fully: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

FACTORIZATION: Grouping in pairs Group terms with common factors or similar brackets! E.g. Group the terms that look similar (i.e. those that could potentially have common factors) Factorize each pair separately and then take out the common bracket:

Switch-arounds “taking out a negative” - x + y = - (x - y) and – x – y = - (x + y) E.g. a) Factorize

b) Factorize: (c) Factorize:

Exercise Factorize: (a) (b) (c) (d) (e)

x³ + y³ = (x+y)(x² - xy + y²) FACTORIZATION: Cubes Sum of Cubes x.x²=x³ y.y²=y³ x³ + y³ = (x+y)(x² - xy + y²) Take the cube root of each term Times factors of first bracket to get middle term Sum of Cubes Example

x³ - y³ = (x-y)(x² + xy + y²) Difference of Cubes x.x²=x³ y.y²=y³ x³ - y³ = (x-y)(x² + xy + y²) Take the cube root of each term Times factors of first bracket to get middle term Difference of Cubes Example Visualization of Factorizing a Cubic Expression

ALGEBRAIC FRACTIONS Simplify the following expressions: (a) (b)

Simplifying Basic Algebraic Expressions Whenever the numerator contains two or more terms, factorize the expression in the numerator and simplify (c) (d) Simplifying Basic Algebraic Expressions

EXERCISE 1 Simplify the following: (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

EXERCISE 2 Simplify (a) (b) (c) (d) (e) (f)

More Advanced Algebraic Fractions Examples a) b) Simplifying Complex Algebraic Expressions

Exercise Simplify: a) b) c) d)