Factor Theorem

C2: Algebra Factor theorem KUS objectives BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem Starter: Expand    

Consider the function f(x) = x2 + 3x - 10 Factorise it: 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) = x2 + 3x - 10 Factorise it: Then: 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). f(x) = (x + 5)(x – 2) f(-5) = 0 and f(2) = 0

Notes 1 If f(p) = 0, then (x – p) is a factor of f(x) f(x)  A function of x, any equation f(p)  The function of x with a value p substituted in For example; Show that (x – 2) is a factor of x3 + x2 – 4x - 4 x3 + x2 – 4x - 4 Substitute in x = 2 23 + 22 – (4x2) - 4 Work out each term 8 + 4 – 8 - 4 = 0 So because f(2) = 0, (x – 2) is a factor of the original equation

(x – 3) is a factor of 𝑓 𝑥 = 𝑥 3 −3 𝑥 2 −9𝑥+27 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). for example (x – 3) is a factor of 𝑓 𝑥 = 𝑥 3 −3 𝑥 2 −9𝑥+27 𝑓 3 = (3) 3 −3 3 2 −9 3 +27=0 Since 𝑥 3 −3 𝑥 2 −9𝑥+27= 𝑥−3 𝑥 2 −9 When x = 3 the bracket = 0 so f(x) = 0

WB 6a If f(p) = 0, then (x – p) is a factor of f(x) f(x)  A function of x, any equation f(p)  The function of x with a value p substituted in For example; Factorise 2x3 + x2 – 18x - 9 2x3 + x2 – 18x - 9 Substitute in values of x to find a factor x = 1 2 + 1 – 18 - 9 = -24 x = 2 16 + 4 – 36 - 9 = -25 x = 3 54 + 9 – 54 - 9 So (x – 3) is a factor = 0

If f(p) = 0, then (x – p) is a factor of f(x) WB 6b If f(p) = 0, then (x – p) is a factor of f(x) f(x)  A function of x, any equation f(p)  The function of x with a value p substituted in For example; Factorise 2x3 + x2 – 18x – 9  Now we know (x – 3) is a factor, divide by it to find the quotient The quotient is 2x2 + 7x + 3 2x2 + 7x + 3 x - 3 2x3 + x2 – 18x - 9 2x3 – 6x2 7x2 – 18x - 9 Third, divide 3x by x = 3 Then, work out 3(x – 3) and subtract from what you have left Second, divide 7x2 by x = 7x Then, work out 7x(x – 3) and subtract from what you have left First, divide 2x3 by x = 2x2 Then, work out 2x2(x – 3) and subtract from what you started with 7x2 – 21x 3x - 9 3x - 9

If f(p) = 0, then (x – p) is a factor of f(x) WB 6c If f(p) = 0, then (x – p) is a factor of f(x) f(x)  A function of x, any equation f(p)  The function of x with a value p substituted in For example; Factorise 2x3 + x2 – 18x – 9 given that f(3) = 0  (x – 3) is a factor  (2x2 + 7x + 3) is the quotient (x – 3)(2x2 + 7x + 3) You can also factorise the quotient 2 numbers that multiply to give +3, and add to give +7 when one has doubled… (x – 3)(2x + 1)(x + 3)

b) Find a given that (𝑥−2) is a factor of 𝑥 3 + 𝑎𝑥 2 – 4𝑥+6. 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 4x4 – 3x2 + a, find the value of a. 4x4 – 3x2 + a If (x + 1) is a factor, then using -1 will make the equation = 0 0 = 4(-14) – 3(-12) + a 0 = 4 – 3 + a Work out each term 0 = 1 + a Solve the equation to find the value of a -1 = a b) Find a given that (𝑥−2) is a factor of 𝑥 3 + 𝑎𝑥 2 – 4𝑥+6. (2)3 + a(2)2 - 4(2) + 6 = 0 4a + 6 = 0 𝒂 =− 𝟑 𝟐

So 𝑥 3 −2 𝑥 2 −8𝑥=(𝑥+2)( 𝑥 2 −4𝑥) 𝑥 2 −4𝑥 𝑥 +2 𝑥 3 −4𝑥 2 +2𝑥 2 −8𝑥 Algebra division: Divide by a bracket using table of values Divided by 𝑥 2 −4𝑥 𝑥 +2 𝑥 3 −4𝑥 2 +2𝑥 2 −8𝑥 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0 Use EWB pen to complete the example So 𝑥 3 −2 𝑥 2 −8𝑥=(𝑥+2)( 𝑥 2 −4𝑥)

So 𝑥 3 +9 𝑥 2 +26𝑥+24=(𝑥+3)( 𝑥 2 +6𝑥+8) 𝑥 2 +6𝑥 +8 𝑥 +3 𝑥 3 +6𝑥 2 +8𝑥 Algebra division: Divide by a bracket using table of values Divided by 𝑥 2 +6𝑥 +8 𝑥 +3 𝑥 3 +6𝑥 2 +8𝑥 +3𝑥 2 +18𝑥 +24 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 0 Use EWB pen to complete the example So 𝑥 3 +9 𝑥 2 +26𝑥+24=(𝑥+3)( 𝑥 2 +6𝑥+8)

1. Show that (x – 2) is a factor of x3 - 4x2 + x + 6. 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 x3 - 4x2 + x + 6. Hence factorise the expression completely. 2. Factorise the expression x3 - 3x2 - 10x + 24. Hence solve the equation x3 - 3x2 - 10x + 24 = 0. 3. If (x – 2) is a factor of f(x) = x3 - 3x2 + a, find the value of a. 4. Determine whether or not (x – 3) is a factor of the expression, x3 - 6x2 + 5x + 12.

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

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 8 Factorise the cubic polynomial f(x) = x3 – 2x2 – x + 2 and hence sketch the graph of the function. Check your answer using Geogebra

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) = x3 + x2 – 5x + 3 as the product of three linear factors. Hence: a) Sketch the graph of the function. b) Solve the equation x3 + x2 – 5x + 3 = 0 Check your answer using Geogebra

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) = x3 + 3x2 – 12x – 14 Hence: a) Sketch the graph of the function. b) Solve f(x) = 0 Check your answer using Geogebra

The polynomial 𝑝(𝑥)is given by 𝑝 𝑥 = 𝑥 3 −19𝑥−30 BAT perform algebraic long division BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem WB11 exam question The polynomial 𝑝(𝑥)is given by 𝑝 𝑥 = 𝑥 3 −19𝑥−30 Use the factor theorem to show that 𝑥+2 is a factor of 𝑝(𝑥) Express 𝑝 𝑥 as the product of three linear factors

factor theorem challenge

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

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