Question

Using the binomial theorem , obtain the expansion of :

Answer 1

Answer:

see explanation

Step-by-step explanation:

Expand both factors and collect like term

Using Pascal' triangle with n = 6 to obtain the coefficients

1  6  15  20  15  6  1

Decreasing powers of 1 from $1^{6}$ to $1^{0}$

Increasing powers of 3x from $(3x)^{0}$ to $(3x)^{6}$

$1+3x)^{6}$

= 1.$1^{6}$$(3x)^{0}$ + 6.$1^{5}$$(3x)^{1}$ + 15.$1^{4}$$(3x)^{2}$ + 20.$1^{3}$$(3x)^{3}$ + 15.1²$(3x)^{4}$ + 6.$1^{1}$$(3x)^{5}$ + 1.$1^{0}$$(3x)^{6}$

= 1 + 18x + 135x² + 540x³ + 1215$x^{4}$ + 1458$x^{5}$ + 729$x^{6}$

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$(1-3x)^{6}$

= 1.$1^{6}$$(-3x)^{0}$ + 6.$1^{5}$$(-3x)^{1}$ + 15.$1^{4}$$(-3x)^{2}$ + 20.$1^{3}$$(-3x)^{3}$ + 15.1²$(-3x)^{4}$ + 6.$1^{1}$$(-3x)^{5}$ + 1.$1^{0}$$(-3x)^{6}$

= 1 - 18x + 135x² - 540x³ + 1215$x^{4}$ - 1458$x^{5}$ + 729$x^{6}$

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Collecting like terms from both expressions

$(1+3x)^{6}$ + $(1-3x)^{6}$

= 2 + 270x² + 2430$x^{4}$ + 1458$x^{6}$

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(2)

Using Pascal's triangle with n = 5

1  5  10  10  5  1

Decreasing powers of 1 from $1^{5}$ to $1^{0}$

Increasing powers of 2x from $(2x)^{0}$ to $(2x)^{5}$

$(1+2x)^{5}$

= 1.$1^{5}$$(2x)^{0}$ + 5.$1^{4}$$(2x)^{1}$ + 10.$1^{3}$$(2x)^{2}$ + 10.$1^{2}$$(2x)^{3}$ + 5.$1^{1}$$(2x)^{4}$+ 1.$1^{0}$$(2x)^{5}$

= 1 + 10x + 40x² + 80x³ + 80$x^{4}$ + 32$x^{5}$