C 2 Algebra Factor theorem KUS BAT BAT

C 2: Algebra Factor theorem KUS BAT BAT objectives perform algebraic long division factorise cubic functions and other expressions recall and use the Factor Theorem Starter: Expand

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem Introduction: Consider the function f(x) = x 2 + 3 x - 10 Factorise it: f(x) = (x + 5)(x – 2) Then: f(-5) = 0 and f(2) = 0 You can now sketch the graph of f(x). The Factor Theorem states that if f(a) = 0 for a polynomial f(x) then (x – a) is a factor of the polynomial f(x).

Notes 1 If f(p) = 0, then (x – p) is a factor of f(x) A function of x, any equation f(p) The function of x with a value p substituted in x 3 + x 2 – 4 x - 4 Substitute in x = 2 23 + 22 – (4 x 2) - 4 Work out each term 8 + 4 – 8 - 4 For example; Show that (x – 2) is a factor of x 3 + x 2 – 4 x - 4 =0 So because f(2) = 0, (x – 2) is a factor of the original equation

Notes 2 The Factor Theorem states that if f(a) = 0 for a polynomial f(x) then (x – a) is a factor of the polynomial f(x). When x = 3 the bracket = 0 so f(x) = 0

WB 6 a If f(p) = 0, then (x – p) is a factor of f(x) A function of x, any equation 2 x 3 + x 2 – 18 x - 9 Substitute in values of x to find a factor f(p) The function of x with a value p substituted in x=1 For example; x=2 = -24 16 + 4 – 36 - 9 = -25 Factorise 2 x 3 + x 2 – 18 x - 9 x=3 So (x – 3) is a factor 2 + 1 – 18 - 9 54 + 9 – 54 - 9 =0

WB 6 b If f(p) = 0, then (x – p) is a factor of f(x) 2 x 2 + 7 x + 3 x-3 f(x) A function of x, any equation f(p) The function of x with a value p substituted in For example; Factorise 2 x 3 + x 2 – 18 x – 9 Now we know (x – 3) is a factor, divide by it to find the quotient The quotient is 2 x 2 + 7 x + 3 2 x 3 + x 2 – 18 x - 9 2 x 3 – 6 x 2 7 x 2 – 18 x - 9 3 by 2 by First, divide Second, Third, divide 2 x 3 x 7 x by xx x 7 x 2 – 21 x 3 x - 9 =3 2 x 2 7 x 3 x - 9 0 Then, work out 3(x 2 x 2(x 7 x(x – – 3) and subtract from whathave you left started have leftwith

WB 6 c If f(p) = 0, then (x – p) is a factor of f(x) (x – 3)(2 x 2 + 7 x + 3) f(x) A function of x, any equation You can also factorise the quotient f(p) The function of x with a value p substituted in 2 numbers that multiply to give +3, and add to give +7 when one has doubled… For example; 2 x 3 x 2 Factorise + – 18 x – 9 given that f(3) = 0 (x – 3) is a factor (2 x 2 + 7 x + 3) is the quotient (x – 3)(2 x + 1)(x + 3)

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB 7 a) Given that (x + 1) is a factor of 4 x 4 – 3 x 2 + a, find the value of a. 4 x 4 – 3 x 2 + a If (x + 1) is a factor, then using -1 will make the equation = 0 0 = 4(-14) – 3(-12) + a Work out each term 0 = 4 – 3 + a Solve the equation to find the value of a 0 = 1 + a -1 = a (2)3 + a(2)2 - 4(2) + 6 = 0 4 a + 6 = 0

Algebra division: Divide by a bracket using table of values Divided by

Algebra division: Divide by a bracket using table of values Divided by

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem Practice 1 1. Show that (x – 2) is a factor of x 3 - 4 x 2 + x + 6. Hence factorise the expression completely. 2. Factorise the expression x 3 - 3 x 2 - 10 x + 24. Hence solve the equation x 3 - 3 x 2 - 10 x + 24 = 0. 3. If (x – 2) is a factor of f(x) = x 3 - 3 x 2 + a, find the value of a. 4. Determine whether or not (x – 3) is a factor of the expression, x 3 - 6 x 2 + 5 x + 12.

Practice 2 Use the factor theorem to show that the following are factors, and hence fully factorise easy harder is a factor of is a factor of is a factor of

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB 8 Factorise the cubic polynomial f(x) = x 3 – 2 x 2 – x + 2 and hence sketch the graph of the function. Check your answer using Geogebra

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB 9 Express f(x) = x 3 + x 2 – 5 x + 3 as the product of three linear factors. Hence: a) Sketch the graph of the function. b) Solve the equation x 3 + x 2 – 5 x + 3 = 0 Check your answer using Geogebra

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB 10: Factorise the cubic polynomial f(x) = x 3 + 3 x 2 – 12 x – 14 Hence: a) Sketch the graph of the function. b) Solve f(x) = 0 Check your answer using Geogebra

BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB 11 exam question

factor theorem challenge

KUS BAT BAT objectives perform algebraic long division factorise cubic functions and other expressions recall and use the Factor Theorem self-assess One thing learned is – One thing to improve is –

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