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Chapter 1 Algebra and functions

Chapter 1 Algebra and functions. C2. Example 1. Simplify this expression 4x 4 +5x 2 -7x x Here write these as three separate fractions. Another example. Simplify the following n 2 +8n +16 n 2 –16

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Chapter 1 Algebra and functions

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  1. Chapter 1 Algebra and functions C2

  2. Example 1 • Simplify this expression • 4x4+5x2-7x • x • Here write these as three separate fractions

  3. 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)

  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

  5. The family division • Remember your family • Dad • Mum • Sister • Brother

  6. 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

  7. No remainder again • Divide 6x3+28x2-7x+15 by (x+5)

  8. Leaving a gap! • Divide x3-3x-2 by (x-2)

  9. 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

  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!

  11. 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

  12. Factor Theorem example 2 • Show the (x-1) is a factor of x3+6x2+5x-12 and hence fully factorize the expression.

  13. 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

  14. 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

  15. 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.

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

  17. 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

  18. 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

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

  20. 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)

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

  22. 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.

  23. Five quick questions • Write five quick quiz questions on the sheet provided

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