Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD PROGRAMME F6 POLYNOMIAL EQUATIONS

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations In Programme F3 a polynomial in the variable x was evaluated by substituting the x-value into the equation and finding the resulting value for the polynomial expression. This process is known as evaluating the expression. Here the process is reversed by giving the polynomial expression the value of zero and finding those values of x which satisfy the resulting equation.

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Quadratic equations, ax 2 + bx + c = 0 Solution by factors Solution by completing the square Solution by formula

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Quadratic equations, ax 2 + bx + c = 0 Solution by factors Where simple factors exist the solution can be derived from those. For example: x 2 + 5x – 14 can be factorized as (x + 7)(x – 2) so if: x 2 + 5x – 14 = 0 then (x + 7)(x – 2) = 0 and so x = −7 or x = 2

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Quadratic equations, ax 2 + bx + c = 0 Solution by completing the square Where simple factors do not exist the solution can be derived from completing the square. For example to solve x 2 – 6x – 4 = 0 it is noted that x 2 – 6x – 4 does not have simple factors so add 4 to both sides to give: x 2 – 6x = 4 Now, add the square of half the x-coefficient to both sides to give: x 2 – 6x + ( – 3) 2 = 4 + (–3) 2 that is x 2 – 6x + 9 = (x – 3) 2 = 13 Therefore x – 3 = ±√13 so x = or x = −0.606 to 3 dp

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Quadratic equations, ax 2 + bx + c = 0 Solution by formula To solve ax 2 + bx + c = 0 use can be made of the formula:

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Solution of cubic equations having at least one linear factor In Programme F3 cubic polynomials were factorized with application of the remainder theorem and the factor theorem and the evaluation of polynomials by nesting. These methods are reapplied to solve cubic equations.

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factors

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Solution of fourth-order equations having at least two linear factors In Programme F3 fourth-order polynomials were factorized with application of the remainder theorem and the factor theorem and the evaluation of polynomials by nesting. These methods are reapplied to solve fourth-order equations.

Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Learning outcomes Solve quadratic equations by factors, completing the square and by formula Solve cubic equations with at least one linear factor Solve fourth-order equations with at least two linear factors