Algebra expansion and simplification

[TwelveMoons]

Hi everyone,

Need some help with two questions please. The first is part of a larger generating functions question and the second is part of a larger recurrence relations question.

First question: How do I expand (1 - x¹⁶)(1 - x¹⁶)(1 - x¹³)(1 - x⁶)

To give: (1 - x⁶ - x¹³ - 2x¹⁶ + x¹⁹ + 2x²² + 2x²⁹ + x³² - x³⁵ + ...

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And second question: How does: \(\displaystyle 3\, =\, A\left(\dfrac{3\, +\, \sqrt{5\,}}{2}\right)\, +\, B\left(\dfrac{3\, -\, \sqrt{5\,}}{2}\right)\)

Become:

\(\displaystyle A\, =\, \left(\dfrac{3\,\sqrt{5\,}\, +\, 5}{10}\right)\)

\(\displaystyle B\, =\, \left(\dfrac{5\, -\, 3\,\sqrt{5\,}}{10}\right)\)

Any help greatly appreciated.

 

[Deleted member 4993]

Hi everyone,

Need some help with two questions please. The first is part of a larger generating functions question and the second is part of a larger recurrence relations question.

First question: How do I expand (1 - x¹⁶)(1 - x¹⁶)(1 - x¹³)(1 - x⁶)

To give: (1 - x⁶ - x¹³ - 2x¹⁶ + x¹⁹ + 2x²² + 2x²⁹ + x³² - x³⁵ + ...

Just multiply it out by using "distributive law" of multiplication i.e.

(a + b) * (c + d) = a*(c + d) + b*(c + d) = a*c + a*d + b*c + b*d

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And second question: How does: \(\displaystyle 3\, =\, A\left(\dfrac{3\, +\, \sqrt{5\,}}{2}\right)\, +\, B\left(\dfrac{3\, -\, \sqrt{5\,}}{2}\right)\)

Become:

\(\displaystyle A\, =\, \left(\dfrac{3\,\sqrt{5\,}\, +\, 5}{10}\right)\)

\(\displaystyle B\, =\, \left(\dfrac{5\, -\, 3\,\sqrt{5\,}}{10}\right)\)

Any help greatly appreciated.

For the second question again expand the right-hand-side by multiplying it out. Then collect the "rational" terms and the "irrational" terms.

If you are still stuck - come back and post your work.

 

[TwelveMoons]

Thanks. I knew the distributive law had to be used just wasn't sure in what order. There was a minor error in the original answer the correct solution is:

Expand: (1 - x¹⁶)(1 - x¹⁶)
Group like terms: 1 - x¹⁶ - x¹⁶ + x³²
Expand: (1 - 2x¹⁶ + x³²)(1 - x¹³)
Group like terms: 1 - 2x¹⁶ + x³² - x¹³ + 2x²⁹ - x⁴⁵
Expand: 1 - 2x¹⁶ + x³² - x¹³ + 2x²⁹ - x⁴⁵(1 - x⁶)
Group like terms and arrange in order: 1 - x⁶ - x¹³ - 2x¹⁶ + x¹⁹ + 2x²² + 2x²⁹ + x³² - 2x³⁵ - x³⁸ - x⁴⁵ + x⁵¹

 

[TwelveMoons]

Thanks. Knew the distributive law had to be used just wasn't sure the order in which to apply it. The original answer has a minor error. Correct answer is:
Expand (1 - x¹⁶)(1 - x¹⁶)
Collect like terms 1 - x¹⁶ - x¹⁶ + x³²
Expand 1 - 2x¹⁶ + x³²(1 - x¹³)
Collect like terms 1 - 2x¹⁶ + x³² - x¹³ + 2x²⁹ - x⁴⁵
Expand 1 - 2x¹⁶ + x³² - x¹³ + 2x²⁹ - x⁴⁵(1 - x⁶)
Collect and order terms 1 - x⁶ - x¹³ - 2x¹⁶ + x¹⁹ + 2x²² + 2x²⁹ + x³² - 2x³⁵ - x³⁸ - x⁴⁵ + x⁵¹