Prerequisite Knowledge
Long/short division.
Dividing algebraic terms.
Factorising quadratic expressions.
Future Links
Factor theorem and remainder theorem.
Prerequisite Knowledge
Long/short division.
Dividing algebraic terms.
Factorising quadratic expressions.
Future Links
Factor theorem and remainder theorem.
Using the Dr Frost online platform
TEACHERS
Generate a random worksheet involving skills in this PowerPoint
(for printing or online task setting).
STUDENTS
Start an independent practice involving skills in this PowerPoint.
drfrost.org/w/782
Clicking this box takes you to a single question practice for a subskill to allow you further Test Your Understanding opportunities. (e.g. drfrost.org/s/123a)
drfrost.org/p/782
Using the Dr Frost online platform
Clicking this box takes you to a single question practice for a subskill to allow you further Test Your Understanding opportunities. (e.g. drfrost.org/s/123a)
Starter
Expand the following (by directly multiplying out the brackets):
Each coefficient is the sum of the two above it.
Pascal’s Triangle
In Pascal’s Triangle, each term (except for the 1s) is the sum of the two terms above.
Tip: We highly recommend memorising each row up to what you see here.
The second number of each row tells us what row we should use for an expansion.
0th row
1st row
2nd row
3rd row
4th row
5th row
Introducing Factorials
1st choice
2nd choice
3rd choice
4th choice
We can choose any of the pieces of fruit for the first slot.
Introducing The Choose Function
🗨 We say as “5 choose 3”
You can avoid using the formula by using the choose button on your calculator.
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Evaluating Factorials & Chooses
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b
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d
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g
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b
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Show all solutions
Example
Simplify each term, ensuring any number in a bracket is raised to the appropriate power.
Example With Negative Term
Test Your Understanding
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Using the Choose Function for Binomial Expansion
0th row
1st row
2nd row
3rd row
Example
The method is the same as before, except you need to calculate the binomial coefficients yourself rather than read them off Pascal’s triangle.
Test Your Understanding
drfrost.org/s/
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I can estimate binomials
Determining a Specific Term of the Expansion
You may want a specific term of an expansion without having to list out unnecessary terms.
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Therefore, what is this exponent?
Due to symmetry in Pascal’s triangle, it doesn’t matter whether we use either of the exponents, 4 or 6.
Test Your Understanding
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drfrost.org/s/
A binomial expansion may be multiplied by another bracket.
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Further Practice
drfrost.org/s/
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We can use a binomial expansion to estimate a power of a number.
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Reflections
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We can use known specific terms to determine an unknown constant.
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Use the ratio given in the question.
Similarly, a ratio may be given between two terms of the expansion.
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Extension Questions
Reflection: This means that the sum of each row in Pascal’s Triangle gives successive powers of 2. Nice!
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I can use the factor and remainder theorem.
Consider the following factorisation of a cubic expression:
Complete the more general statement…
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Examples
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b
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For any ‘show that’ question, you must have a conclusion.
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Test Your Understanding
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Determining Roots Using the Factor Theorem
Factorising a Cubic When a Factor is Known
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You may be required to find multiple factors to fully factorise a cubic.
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We can use the factor theorem to determine an unknown coefficient.
By the factor theorem.
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Harder questions may involve two unknown coefficients, so we need two equations.
We are given a second factor this time.
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The Remainder Theorem
We can extend the factor theorem to quickly determine the remainder from an algebraic division.
Recall how we can rewrite a division in terms of the quotient and remainder:
Quotient
Remainder
Divisor
Quotient
Remainder
Divisor
% Please check your exam specification to see if this topic is covered.
The Remainder Theorem
Quotient
Remainder
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By the remainder theorem.
Just like with the factor theorem, we can use the remainder theorem to determine unknown coefficients in a polynomial.
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By the remainder theorem.
If there are two unknowns, we need two pieces of information about factors and/or remainders.
By the factor theorem.
Substituting.
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