Chapter 1 Algebra and Chapter 1 Algebra and functions functions

C2 C2

Example 1 Example 1

Simplify this expression Simplify this expression 4x4x44+5x+5x22-7x-7x x x Here write these as three separate Here write these as three separate

fractionsfractions

Another exampleAnother example

Simplify the following Simplify the following nn22+8n +16+8n +16 nn22 –16 –16 This is a perfect square and a difference of This is a perfect square and a difference of

two squares- cookie cutters!two squares- cookie cutters! == (n+4) (n+4)2 2 (n-4)(n+4)(n-4)(n+4) = = (n+4)(n+4) (n-4)(n-4)

Dividing polynomials by Dividing polynomials by cancellingcancelling

Always factorise first using the HCF Always factorise first using the HCF and then cancel out common factors and then cancel out common factors in the fractionin the fraction

Example Example (2c(2c22dd33))22

8b8b44cc44

Quiz Quiz Some classwork questionsSome classwork questions

The family division The family division

Remember your familyRemember your family DadDad MumMum SisterSister BrotherBrother

Example 1 No remainder Example 1 No remainder

Set you work out using the family Set you work out using the family stepssteps

Divide xDivide x33 + 2 x + 2 x22 -17x +6 by (x-3) -17x +6 by (x-3)

x-3 ) xx-3 ) x33+2x+2x22–17x +6–17x +6

No remainder againNo remainder again

Divide 6xDivide 6x33+28x+28x22-7x+15 by (x+5) -7x+15 by (x+5)

Leaving a gap!Leaving a gap!

Divide xDivide x33-3x-2 by (x-2) -3x-2 by (x-2)

Example with a remainderExample with a remainder

Divide 2xDivide 2x33–5x–5x22-16x +10 by (x-4)-16x +10 by (x-4) Now we must write the polynomial Now we must write the polynomial

with highest power first!with highest power first!

x –4 ) 2xx –4 ) 2x33 -5x -5x22 –16x+ 10 –16x+ 10

A summary of long division A summary of long division

Always follow Dad, mum, sister, Always follow Dad, mum, sister, brotherbrother

Always write the polynomial in order Always write the polynomial in order starting with the highest powerstarting with the highest power

Always leave a space for any terms Always leave a space for any terms (powers) not in the question(powers) not in the question

More worked examples?More worked examples? Click on the picture!Click on the picture!

Factor theorem Factor theorem

if x-a is a factor of f(x) then if x-a is a factor of f(x) then f(a)=0f(a)=0

What does this mean?What does this mean? If (x-2) is a factor of If (x-2) is a factor of f(x) = xf(x) = x33+x+x22-4x-4 then f(2) = 0. -4x-4 then f(2) = 0.

please check this. please check this. F(2) = 8 +4-8-4F(2) = 8 +4-8-4

Factor Theorem example 2Factor Theorem example 2

Show the (x-1) is a factor of Show the (x-1) is a factor of xx33+6x+6x22+5x-12 and hence fully +5x-12 and hence fully factorize the expression. factorize the expression.

Example 3 Example 3

Given that (x+1) is a factor of Given that (x+1) is a factor of

4x4x44-3x-3x22+a find the value of a. +a find the value of a. F(-1) = 4-3+aF(-1) = 4-3+a 1 + a = 01 + a = 0 a = -1 a = -1

Example 4Example 4

Prove that (2x+1) is a factor of Prove that (2x+1) is a factor of 2x2x33+x+x22-18x-9-18x-9

What do you substitute here? What do you substitute here? Here we substitute x = -1/2 Here we substitute x = -1/2 F(-1/2) = 2 (-1/8) + ¼ +9-9F(-1/2) = 2 (-1/8) + ¼ +9-9 = 0 = 0

Finding factors of a polynomialFinding factors of a polynomial Fully factorise f(x) = 2xFully factorise f(x) = 2x33+x+x22-18x-9-18x-9 The first step here is to look at the constant 9 and The first step here is to look at the constant 9 and

try the factors of 9.try the factors of 9. We will try f(1), f(-1), f( 3) and see which one We will try f(1), f(-1), f( 3) and see which one

equals 0.equals 0. F(1) = 2+1-18-9F(1) = 2+1-18-9 F(3) = 54 + 9 -54 – 9!F(3) = 54 + 9 -54 – 9! So (x-3) is a factor.So (x-3) is a factor. Now we use family division to find the other Now we use family division to find the other

factors. factors.

Ex 1D. Ex 1D.

Classwork Classwork

Just one more practice question on Just one more practice question on Sos maths. Let’s do this together.Sos maths. Let’s do this together.

Remainder Theorem Remainder Theorem

If when you substitute f(a) into the If when you substitute f(a) into the polynomial and it does not equal zero polynomial and it does not equal zero then this number is actually the then this number is actually the remainder. remainder.

Example find the remainder when Example find the remainder when xx33-20x+3 is divided by (x-4) -20x+3 is divided by (x-4)

F(4) = -13 F(4) = -13 Ex 1E Mixed exercise 1F Ex 1E Mixed exercise 1F

A competition A competition

1) Show that x-2 is a factor of1) Show that x-2 is a factor of f(x) = xf(x) = x33+x+x22-5x-2 and hence or -5x-2 and hence or

otherwise find the exact solutions of otherwise find the exact solutions of the equation f(x) = 0 the equation f(x) = 0

A competition A competition

2) Given that x = -1 is a root of the 2) Given that x = -1 is a root of the equation 2xequation 2x33-5x-5x22-4x+3, find the other -4x+3, find the other two positive roots.two positive roots.

A competition A competition

3) H(x) = x3) H(x) = x33+4x+4x22+rx+s. Given that+rx+s. Given that H(-1) = 0 and H(2) = 30, find the H(-1) = 0 and H(2) = 30, find the

values of r and s. Find the remainder values of r and s. Find the remainder when H(x) is divided by (3x-1)when H(x) is divided by (3x-1)

A competition A competition

4) Given that g(x) = 2x4) Given that g(x) = 2x33+9x+9x22-6x-5, -6x-5, factorise g(x) and solve g(x) = 0 .factorise g(x) and solve g(x) = 0 .

A competition A competition

5) F(x) = 2x5) F(x) = 2x22+px+q. Given that+px+q. Given that

F(-3)=0 and F(4) = 2 find the value of F(-3)=0 and F(4) = 2 find the value of p and q and hence fully factorise. p and q and hence fully factorise.

Five quick questionsFive quick questions

Write five quick quiz questions on the Write five quick quiz questions on the sheet providedsheet provided