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Chapter 1 Algebra and functions PowerPoint Presentation

Chapter 1 Algebra and functions

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## PowerPoint Slideshow about ' Chapter 1 Algebra and functions ' - gillian-vincent

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Example 1

- Simplify this expression
- 4x4+5x2-7x
- x
- Here write these as three separate fractions

Another example

- Simplify the following n2+8n +16
- n2 –16
- This is a perfect square and a difference of two squares- cookie cutters!
- = (n+4)2
- (n-4)(n+4)
- = (n+4)
- (n-4)

Dividing polynomials by cancelling

- Always factorise first using the HCF and then cancel out common factors in the fraction
- Example (2c2d3)2
- 8b4c4
- Quiz
- Some classwork questions

The family division

- Remember your family
- Dad
- Mum
- Sister
- Brother

Example 1 No remainder

- Set you work out using the family steps
- Divide x3 + 2 x2 -17x +6 by (x-3)
- x-3 ) x3+2x2–17x +6

No remainder again

- Divide 6x3+28x2-7x+15 by (x+5)

Leaving a gap!

- Divide x3-3x-2 by (x-2)

Example with a remainder

- Divide 2x3–5x2-16x +10 by (x-4)
- Now we must write the polynomial with highest power first!
- x –4 ) 2x3 -5x2 –16x+ 10

A summary of long division

- Always follow Dad, mum, sister, brother
- Always write the polynomial in order starting with the highest power
- Always leave a space for any terms (powers) not in the question
- More worked examples?
- Click on the picture!

Factor theorem

- if x-a is a factor of f(x) then f(a)=0
- What does this mean?
- If (x-2) is a factor of
- f(x) = x3+x2-4x-4 then f(2) = 0. please check this.
- F(2) = 8 +4-8-4

Factor Theorem example 2

- Show the (x-1) is a factor of x3+6x2+5x-12 and hence fully factorize the expression.

Example 3

- Given that (x+1) is a factor of
4x4-3x2+a find the value of a.

- F(-1) = 4-3+a
- 1 + a = 0
- a = -1

Example 4

- Prove that (2x+1) is a factor of 2x3+x2-18x-9
- What do you substitute here?
- Here we substitute x = -1/2
- F(-1/2) = 2 (-1/8) + ¼ +9-9
- = 0

Finding factors of a polynomial

- Fully factorise f(x) = 2x3+x2-18x-9
- The first step here is to look at the constant 9 and try the factors of 9.
- We will try f(1), f(-1), f( 3) and see which one equals 0.
- F(1) = 2+1-18-9
- F(3) = 54 + 9 -54 – 9!
- So (x-3) is a factor.
- Now we use family division to find the other factors.
- Ex 1D.

Classwork

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

Remainder Theorem

- If when you substitute f(a) into the polynomial and it does not equal zero then this number is actually the remainder.
- Example find the remainder when x3-20x+3 is divided by (x-4)
- F(4) = -13
- Ex 1E Mixed exercise 1F

A competition

- 1) Show that x-2 is a factor of
- f(x) = x3+x2-5x-2 and hence or otherwise find the exact solutions of the equation f(x) = 0

A competition

- 2) Given that x = -1 is a root of the equation 2x3-5x2-4x+3, find the other two positive roots.

A competition

- 3) H(x) = x3+4x2+rx+s. Given that
- H(-1) = 0 and H(2) = 30, find the values of r and s. Find the remainder when H(x) is divided by (3x-1)

A competition

- 4) Given that g(x) = 2x3+9x2-6x-5, factorise g(x) and solve g(x) = 0 .

A competition

- 5) F(x) = 2x2+px+q. Given that
F(-3)=0 and F(4) = 2 find the value of p and q and hence fully factorise.

Five quick questions

- Write five quick quiz questions on the sheet provided

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