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Chapter 7: Problem 2

If \((1-p)\left(1+3 x+9 x^{2}+27 x^{3}+81 x^{4}+243 x^{5}\right)\) \(=1-\mathrm{p}^{6}, \mathrm{p} \neq 1\), then the value of \(\frac{\mathrm{p}}{\mathrm{x}}\) may be equal to (1) \(\frac{1}{3}\) (2) 3 (3) \(\frac{1}{2}\) (4) 2

Short Answer

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Step by step solution

01

Rewrite the Given Equation

Consider the given equation equation: \((1-p)(1+3x+9x^{2}+27x^{3}+81x^{4}+243x^{5})=1-p^{6}\) .
02

Recognize the Geometric Series

Observe that the term \(1+3x+9x^{2}+27x^{3}+81x^{4}+243x^{5}\) is a geometric series with the first term as 1 and common ratio as 3x.
03

Sum of the Geometric Series

The sum of a geometric series \(1 + r + r^2 + ... + r^n\) is given by \( \frac{r^{n+1} - 1}{r-1} \). Here, the common ratio \( r = 3x \) and the series has 6 terms. So, the sum is equation: \( \frac{(3x)^6 - 1}{3x - 1} \).
04

Substitute and Simplify

Substitute this sum back into the equation: equation: \((1-p) \cdot \frac{(3x)^6 - 1}{3x - 1}= 1 - p^6\) .
05

Factorize and Solve

Simplifying the equation by cross-multiplying, we get: equation: \((3x)^6 - 1 = (3x - 1)(1 - p^6)\).Continuing this step, comparing coefficients will yield the relationship between \(p\) and \(x\).
06

Compare Terms

For both sides to be equal, \((3x)^6 = p^6\). This implies equation: \(3x = p\) or \(3x = -p\).
07

Solve for p/x

\(p = 3x\) or \(p = -3x\).Thus, \(\frac{p}{x} = 3\) or \(\frac{p}{x} = -3\). Since \(\frac{p}{x}\) is clearly a positive term, \(\frac{p}{x}= 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this problem, the given series is:
  • 1 + 3x + 9x² + 27x³ + 81x⁴ + 243x⁵
Here, the first term is 1, and the common ratio is 3x. The concept of geometric series is useful for summing up the entire sequence using a simple formula rather than adding each term individually. This becomes handy in algebraic manipulation and solving polynomial equations, as seen in this exercise.
Polynomial Equations
Polynomial equations involve expressions with multiple terms, where variables are raised to whole-number exponents and multiplied by coefficients.
In our exercise, we deal with the polynomial:
  • (1−p)(1+3x+9x²+27x³+81x⁴+243x⁵) = 1−p6
Polynomial equations often appear in algebra, and solving them involves finding the values of the unknowns that satisfy the given equation. This forms the basis of mathematics used in various scientific and engineering fields.
Algebraic Manipulation
Algebraic manipulation involves rewriting expressions and equations in different forms to simplify or solve them. This includes factoring, expanding, and rearranging terms.
In our problem, we recognize the geometric series and use the sum formula:
  • sum of series = [(3x)6−1]/(3x−1)
We then substitute this back into the equation:
  • (1−p) [ (3x)^6−1 ]/(3x−1) = 1−p6
From here, we simplify the algebraic expression through cross-multiplication to compare terms accurately.
Equation Solving
Equation solving is the process of finding the values of variables that make the equation true. It's a fundamental skill in algebra. In this exercise, after manipulating the polynomial equation, we set:
  • (3x)6 = p6
    • .
This comparison lets us deduce two possible relationships:
  • p=3x
  • or p=−3x
But since p/x must be positive, we find p=3x. Hence the ratio p/x=3. By understanding each step and concept, grasping how to solve such equations becomes much simpler.
Breaking down complex problems into individual concepts aids in better comprehension.

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