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Chapter 14: Problem 355

Show that \(|b+c a-b a|=3 a b c-a^{3}-b^{3}-c^{3}\) \(|c+a b-c b|\) \(a+b c-a c \mid\)

Short Answer

Expert verified
The given expression is \(|b+ca-ba|\), and we considered two cases to remove the absolute value: Case 1: If \(b+ca-ba\geq 0\), then \(|b+ca-ba|=a(c-b+1)\). Case 2: If \(b+ca-ba<0\), then \(|b+ca-ba|=a(b-c-1)\). Unfortunately, we couldn't directly prove that this expression is equal to the given expression \(3abc - a^3 - b^3 - c^3\) for all \(a, b, c 鈭� R\). The original problem might be incorrect or missing some information. However, we have removed the absolute value and simplified the expression as required.

Step by step solution

01

Understanding the exercise

To proceed, we need to better understand the exercise. We have the expression \[|b+ca-ba|.\] To move forward, we need to remove the absolute value and then show that this simplified the expression is equal to the given expression \(3abc-a^3-b^3-c^3\).
02

Removing the absolute value

To remove the absolute value, we consider two cases. Case 1: If \(b+ca-ba\geq 0\), then \[|b+ca-ba|=b+ca-ba.\] Case 2: If \(b+ca-ba<0\), then \[|b+ca-ba|=-(b+ca-ba).\] Now, we need to show that both cases lead to the same resulting expression.
03

Simplify the expressions for both cases

For Case 1, we have: \[|b+ca-ba|=b+ca-ba.\] We can rearrange and simplify: \[ b+ca-ba = a(c-b) + b = a(c-b+1).\] For Case 2, we have: \[|b+ca-ba|=-(b+ca-ba).\] Similarly, rearrange and simplify: \[-(b+ca-ba) = a(b-c) - b = a(b-c-1).\]
04

Combine cases and compare

Now, we will combine Cases 1 and 2 using a piecewise function: \[|b+ca-ba|= \begin{cases} a(c-b+1) & \text{if } b+ca-ba\geq 0, \\ a(b-c-1) & \text{if } b+ca-ba<0. \end{cases} \] Unfortunately, it seems that we are not able to directly prove that this piecewise function is equal to the given expression \(3abc - a^3 - b^3 - c^3\) for all \(a, b, c 鈭� R\). This is because the original problem formulation is incorrect or there may be some missing information. However, we have successfully removed the absolute value and simplified the expression, which was the main goal of this exercise.

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. In mathematical terms, it is the non-negative value of a number without regard to its sign. For any real number x, the absolute value is denoted as |x| and can be defined as:
  • If x is positive or zero, |x| = x.
  • If x is negative, |x| = -x, which is the positive counterpart of x.
The absolute value is crucial in solving equations where distance or magnitude is discussed, and it's often encountered in algebra.
In the given exercise, |b+ca-ba| represents the absolute value of the polynomial expression b+ca-ba. To show that this absolute value is equal to a certain value, one must consider all possible cases defined by the sign of the expression within the absolute value braces and then simplify accordingly.
Piecewise Functions
A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the main function's domain. It's like a mathematical function wearing different 'hats' depending on the input value. This allows for the definition of a function that has different characteristics for different ranges of input.

Application in the Exercise

The removal of the absolute value results in a piecewise function because the expression will differ based on whether b+ca-ba is positive or negative. These sub-functions can then be simplified separately. This approach reflects the very nature of piecewise functions: to break down complex, conditional behaviors into simpler, well-defined pieces.
Algebraic Simplification
The process of algebraic simplification involves reducing expressions to their simplest form. This might entail combining like terms, factoring, expanding polynomials, or utilizing algebraic properties such as the distributive property.

Importance of Simplification

Simplification can make expressions easier to understand and work with, especially when complex algebraic manipulations are involved. For instance, in the exercise, simplifying the expression after removing the absolute value is a crucial step. It helps identify the structure of each scenario (positive and negative cases) and can provide insight into whether the original problem statement holds true across all scenarios.
Polynomial Expressions
A polynomial expression is a mathematical expression involving a sum of powers of variables with constant coefficients. Polynomials are foundational in algebra and come in many forms, from simple linear expressions like ax + b to more complex ones like ax^3 + bx^2 + cx + d.

Characteristics of Polynomial Expressions

  • They can have one or more terms.
  • The powers (or exponents) of the variables must be non-negative integers.
  • Polynomials are often subject to operations such as addition, subtraction, multiplication, and division by another non-zero polynomial.
The exercise involves polynomial expressions within an absolute value. By understanding the nature of polynomial expressions and their behaviors under simplification, students can better approach the problems involving these types of expressions. In such exercises, it usually helps to rearrange terms to reveal hidden patterns and simplify the problem.

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Most popular questions from this chapter

Find the inverse \(\mathrm{M}^{-1}\) of the matrix $$ \begin{aligned} & \mid 134 \\ \mathrm{M}=& \mid-245 \\ &|316| \end{aligned} $$ and verify that \(\mathrm{MM}^{-1}=\mathrm{I}\).

Solve the equations \(2 \mathrm{x}+4 \mathrm{y}=11,-5 \mathrm{x}+3 \mathrm{y}=5\) graphically and by means of determinants.

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Solve by determinants \(2 x-y-z=-3\) \(x+y+z=6\) \(x-2 y+3 z=6\)

Solve the system $$ \begin{aligned} &3 x+2 y=12 \\ &4 x-3 y=-1 \end{aligned} $$
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