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Chapter 6: Problem 76

Solve the given problems. Is it true that \((1-2 x)^{3}=(2 x-1)^{2}(1-2 x) ?\)

Short Answer

Expert verified
Yes, the equation is true; both sides are equal after expansion.

Step by step solution

01

Expand the Left-Hand Side (LHS)

Begin by expanding \((1-2x)^3\). This can be expanded using the binomial theorem, giving us:\[(1-2x)^3 = 1 - 3(2x) + 3(2x)^2 - (2x)^3 = 1 - 6x + 12x^2 - 8x^3\].The expanded form of the LHS is \(1 - 6x + 12x^2 - 8x^3\).
02

Expand the Right-Hand Side (RHS)

Now, expand \((2x-1)^2(1-2x)\).First, find \((2x-1)^2\):\[(2x-1)^2 = (2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1\].Next, multiply this by \((1-2x)\):\[(4x^2 - 4x + 1)(1-2x) = 4x^2(1-2x) - 4x(1-2x) + 1(1-2x).\]Expanding each term separately:1. \(4x^2(1-2x) = 4x^2 - 8x^3\).2. \(-4x(1-2x) = -4x + 8x^2\).3. \(1(1-2x) = 1 - 2x\).Combine these results:\[4x^2 - 8x^3 - 4x + 8x^2 + 1 - 2x = -8x^3 + 12x^2 - 6x + 1\].This is the expanded form of the RHS.
03

Compare Both Sides

Compare the fully expanded forms of both sides of the equation:LHS: \(1 - 6x + 12x^2 - 8x^3\)RHS: \(-8x^3 + 12x^2 - 6x + 1\).Since the expanded expressions for LHS and RHS are identical, the initial equation holds true.

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Key Concepts

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

Polynomial Expansion
Polynomial expansion is the process of expressing a polynomial raised to a power in its expanded form. This involves distributing the powers across the terms and combining like terms. In our exercise, we encountered the polynomial \((1 - 2x)^3\). Using the binomial theorem, we expanded this polynomial into its full form: \(1 - 6x + 12x^2 - 8x^3\). The binomial theorem helps in expanding expressions of the form \((a + b)^n\), where you apply specific coefficients (from Pascal's triangle) to power each term appropriately and sum them up.
Equation Solving
Equation solving involves manipulating an equation to find an unknown value or to prove its validity. In the exercise provided, we needed to establish whether two sides of an equation were equal. First, we expanded each side of the equation separately.
  • Left-hand side (LHS): \( (1 - 2x)^3 \) expanded to \( 1 - 6x + 12x^2 - 8x^3 \)
  • Right-hand side (RHS): \((2x - 1)^2(1 - 2x)\) expanded to \(-8x^3 + 12x^2 - 6x + 1\)
Both equations evaluated to the same expression upon expansion, proving the equation true. This method of solving a problem typically involves several steps, including expansion, simplification, and comparison.
Algebraic Expressions
Algebraic expressions consist of variables, constants, and operations such as addition, subtraction, multiplication, and division. In the context of the exercise, we worked with algebraic expressions that involved terms like \(2x\), \(-1\), and powers of \(x\) such as \(x^2\) and \(x^3\). By expanding these expressions, we combined like terms and simplified the complex equations. This process is fundamental in algebra as it allows us to break down and solve equations systematically.
Understanding how to manipulate and work with these expressions is key to successfully solving algebraic problems.

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