1. STROUD Worked examples and exercises are in the text 1 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations PROGRAMME F3 EXPRESSIONS and EQUATIONS

2. STROUD Worked examples and exercises are in the text 2 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

3. STROUD Worked examples and exercises are in the text 3 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

4. STROUD Worked examples and exercises are in the text 4 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Evaluating expressions Equations Evaluating independent variables Transposition of formulas

5. STROUD Worked examples and exercises are in the text 5 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Evaluating expressions When numerical values are assigned to the variables and constants in an algebraic expression, the expression itself assumes a numerical value that is obtained by following the usual precedence rules. This process is known as evaluating the expression.

6. STROUD Worked examples and exercises are in the text 6 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Because different values of the variables in an algebraic expression (called the independent variables) produce different values for the expression, we assign these expression values to another variable (called the dependent variable) and so form an equation. There are different types of equation.

7. STROUD Worked examples and exercises are in the text 7 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Conditional equation A conditional equation, usually just called an equation, is true only for certain values of the symbols involved. For example:

8. STROUD Worked examples and exercises are in the text 8 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Identity An identity is a statement of the equality of two expressions that is true for all values of the symbols for which both expressions are defined..

9. STROUD Worked examples and exercises are in the text 9 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Defining equation A defining equation is a statement of equality that defines an expression.

10. STROUD Worked examples and exercises are in the text 10 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Assigning equation An assigning equation is a statement of equality that assigns a specific value to a variable.

11. STROUD Worked examples and exercises are in the text 11 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Equations Formula A formula is a statement of equality that expresses a mathematical fact where all the variables, dependent and independent, are well-defined.

12. STROUD Worked examples and exercises are in the text 12 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Evaluating independent variables Sometimes the numerical values assigned to the variables and constants in a formula include a value of the dependent variable and exclude a value of one of the independent variables. The exercise is then to find the corresponding value of that independent variable by transposing the equation. For example:

13. STROUD Worked examples and exercises are in the text 13 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations Transposition of formulas A formula can be transposed even when values for the variables and constants have not been assigned. For example:

14. STROUD Worked examples and exercises are in the text 14 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

15. STROUD Worked examples and exercises are in the text 15 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

16. STROUD Worked examples and exercises are in the text 16 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations The evaluation process Systems A system is a process that is capable of accepting an input, processing the input and producing an output. We can use this idea of a system to describe the way we evaluate an algebraic expression. fx f(x) We can describe the evaluation of 3x – 4 when x = 5 as f processing the input 5 to produce the output 11. The output f(x) is then the result of: ‘ f acting on x ’

17. STROUD Worked examples and exercises are in the text 17 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

18. STROUD Worked examples and exercises are in the text 18 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

19. STROUD Worked examples and exercises are in the text 19 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Polynomial equations Polynomial expressions Evaluation of a polynomial by nesting

20. STROUD Worked examples and exercises are in the text 20 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Polynomial equations Polynomial expressions A polynomial in x is an expression involving powers of x, normally arranged in descending (or sometimes ascending) powers. The degree of the polynomial is given by the highest power of x occurring in the expression. For example:

21. STROUD Worked examples and exercises are in the text 21 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Polynomial equations Evaluation of a polynomial by nesting To express a polynomial in nested form write down the coefficient and one factor x from the first term and add on the coefficient of the next term. Enclose in brackets, multiply by x and add on the next coefficient. Repeat the process. For example: This simplifies the evaluation process. For instance:

22. STROUD Worked examples and exercises are in the text 22 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

23. STROUD Worked examples and exercises are in the text 23 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

24. STROUD Worked examples and exercises are in the text 24 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Remainder theorem The remainder theorem states that if a polynomial f(x) is divided by (x – a), the quotient will be a polynomial g(x) of one degree less than the degree of f(x), together with a remainder R still to be divided by (x – a). That is:

25. STROUD Worked examples and exercises are in the text 25 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Remainder theorem Factor theorem If f(x) is a polynomial and substituting x = a gives a zero remainder, that is f(a) = 0, then (x – a) is a factor of f(x).

26. STROUD Worked examples and exercises are in the text 26 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

27. STROUD Worked examples and exercises are in the text 27 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

28. STROUD Worked examples and exercises are in the text 28 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Factorization of cubic polynomials Given the cubic f(x) find the value of a by trial and error for which f(a) = 0 so that (x – a) is a factor of f(x). Divide f(x) by x – a to obtain a quadratic which can then be factorized into its two factors. For example: From this it can be seen that f(3) = 0 so (x – 3) is a factor of f(x). Division produces the quadratic x 2 – 2x – 8 = (x – 4)(x + 2) and so:

29. STROUD Worked examples and exercises are in the text 29 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

30. STROUD Worked examples and exercises are in the text 30 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Expressions and equations The evaluation process Polynomial equations Remainder theorem Factorization of cubic polynomials Factorization of fourth-order polynomials

31. STROUD Worked examples and exercises are in the text 31 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Factorization of fourth-order polynomials Given the fourth-order f(x) find the value of a by trial and error for which f(a) = 0 so that (x – a) is a factor of f(x). Divide f(x) by x – a to obtain a cubic which can then be factorized into its three factors as done previously.

32. STROUD Worked examples and exercises are in the text 32 STROUD Worked examples and exercises are in the text Programme F3: Expressions and equations Learning outcomes Numerically evaluate an algebraic expression by substituting numbers for variables Recognize the different types of equation Evaluate an independent variable Change the subject of an equation by transposition Evaluate polynomial expressions by ‘ nesting ’ Use the remainder and factor theorems to factorize polynomials Factorize cubic polynomials Factorize fourth-order polynomials