Pascal's triangle

Pascal's triangle

A Pascal's triangle is a number triangle of the binomial coefficients. The first row of the triangle is always 1.
Basic concepts: Multiplying binomial by binomial, Combinations,

Lessons

  • 1.
    Expand:
    a)
    i) (a+b)0{\left( {a + b} \right)^0} =
    ii) (a+b)1{\left( {a + b} \right)^1} =
    iii) (a+b)2{\left( {a + b} \right)^2} =
    iv) (a+b)3{\left( {a + b} \right)^3} =

    Use Pascal’s Triangle to expand:
    i) (a+b)4{\left( {a + b} \right)^4} =
    ii) (a+b)5{\left( {a + b} \right)^5} =

    b)
    Investigating Pascal’s Triangle

  • 2.
    Pascal’s Triangle - sum of numbers in each row
    a)
    Row Pattern Corresponding binomial expression Sum of the numbers in,the row Express the sum as a power of 2
    1 1 (a+b)0{\left( {a + b} \right)^0}
    2 1 1 (a+b)1{\left( {a + b} \right)^1}
    3 1 2 1 (a+b)2{\left( {a + b} \right)^2}
    4 1 3 3 1 (a+b)3{\left( {a + b} \right)^3}
    5 1 4 6 4 1 (a+b)4{\left( {a + b} \right)^4}
    : : : : :
    n (a+b)n−1{\left( {a + b} \right)^{n - 1}}
    n+1 (a+b)n{\left( {a + b} \right)^n}

    b)
    What is the sum of the numbers in the 10th row of Pascal’s Triangle?

    c)
    What is the sum of the coefficients in the expansion of (a+b)50{\left( {a + b} \right)^{50}} ?

    d)
    Express the number pattern of Pascal’s triangle in "combination" form, then deduce the following formula:
    \; nC0{}_n^{}{C_0} + nC1{}_n^{}{C_1} + nC2{}_n^{}{C_2} + nC3{}_n^{}{C_3} + … + nCn−2{}_n^{}{C_{n - 2}} + nCn−1{}_n^{}{C_{n - 1}} + nCn{}_n^{}{C_n} = 2n{2^n}

  • 3.
    Without using a calculator, evaluate:
    a)
    10C0{}_{10}^{}{C_0} + 10C1{}_{10}^{}{C_1} + 10C2{}_{10}^{}{C_2} + … + 10C9{}_{10}^{}{C_9} + 10C10{}_{10}^{}{C_{10}}

    b)
    5C0{}_5^{}{C_0} + 5C1{}_5^{}{C_1} + 5C2{}_5^{}{C_2} + 5C3{}_5^{}{C_3} + 5C4{}_5^{}{C_4}

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2.
Polynomials
2.1
Polynomial long division
2.2
Polynomial synthetic division
2.3
Determining the equation of a polynomial function
2.4
Factor theorem
2.5
Rational zero theorem
2.6
Characteristics of polynomial graphs
2.7
Multiplicities of polynomials
2.8
Imaginary zeros of polynomials
2.9
Pascal's triangle
2.10
Binomial theorem
2.11
What is a polynomial function?
2.12
Applications of polynomial functions
2.13
Solving polynomial inequalities
2.14
Descartes’ rule of signs
Precalculus
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