Fichas
Expanding and factorising
Fichas del conjunto (12)
- 1Explain the method to expand a cubic such as : (x+2)(x+3)(x+4)Expand 2 brackets into a quadratic and then multiply the quadratic by the final bracket
- 2Explain the method to expand a cubic such as : x(x+3)^2Write (x+3)^2 as (x+3)(x+3) and expand, and then multiply the answer by the x outside the brackets
- 3Explain the method to expand a quartic such as: (x-1)^2(x+1)^2Rewrite as ((x-1)(x+1))^2, which is the same as (x^2-1)^2, and now multiply this out
- 4Explain a method to factorise a cubic such as: x^3+6x^2+9xX is a common factor in this particular example, so factor out x giving x(x^2+6x+9) and then factorise the quadratic inside the brackets, leaving the x outside
- 5Explain the method to factorise a quartic such as : x^4-2x^2+1Rewrite as the quadratic: (x^2)^2-2(x^2)+1 , saying (x^2) = x. So x^2-2x+1 = (x-1)(x-1). Now as x=(x^2) we rewrite this as (x^2-1)(x^2-1) which = (x+1)^2(x-1)^2 , so (x+1)(x+1)(x-1)(x-1)
- 6What is the factor theorem?For a polynomial f(x), if f(a)=0 then (x-a) is a factor
(e.g. f(x)=x^2-3x-10 , f(5)=0, so (x-5) is a factor) - 7In an exam, how would we construct an answer (or part of an answer) including the factor theorem?f(a)=0 , so by the factor theorem (x-a) is a factor of f(x)
- 8In 3 steps, describe how to factorise a cubic.1. Factor Theorem
2. Split f(x) into linear term and a quadratic
3. Use box method (or comparison method) to find a,b and c of the quadratic which you can then solve - 9What simple trick can check if you have worked out a cubics roots correctly?So ax^3+bx^2+cx+d
The factors will multiply to make d and add to make b - 10What is the remainder theorem?For the polynomial f(x), if f(a)=b then the remainder when f(x) / (x-a) is b
- 11How would we find the remainder of a quadratic when it's divided by (x-1)?Use the function f(1) and collect together the terms
- 12How would we find a when f(x)=x^3-2x^2+ax+3 is divided by (x+1) to give the remainder 5?Use function f(-1) to get -1-2-a+3=5 , then rearrange for a giving a=-5