The Polynomials P1 = 2x^2+1, P2= - 4x^2+x And P3 =x-2 Are Linearly Dependent. Select One: True False

The Polynomials P1 = 2x^2+1, P2= - 4x^2+x And P3 =x-2 Are Linearly Dependent. Select One: True False

In the study of polynomials within linear algebra, understanding the concepts of linear dependence and independence is fundamental. When working with polynomial sets, such as P1 = 2x^2 + 1, P2 = -4x^2 + x, and P3 = x - 2, determining whether these polynomials are linearly dependent or independent helps in understanding their span, basis, and the structure of polynomial vector spaces. In this article, we will explore the concept of linear dependence among these specific polynomials, how to verify it, and the implications of such dependence in mathematical contexts.

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Understanding Linear Dependence and Independence of Polynomials

What Is Linear Dependence?

Linear dependence among a set of vectors (or polynomials) exists when at least one element of the set can be expressed as a linear combination of the others. In the context of polynomials, this means there exist scalars, not all zero, such that: \[ c1 P1 + c2 P2 + c3 P3 = 0 \] where \( c1, c2, c_3 \) are real numbers, and 0 denotes the zero polynomial.

What Is Linear Independence?

Conversely, polynomials are linearly independent if the only solution to the above equation is when all scalars are zero: \[ c1 = c2 = c_3 = 0 \] This indicates no polynomial in the set can be written as a combination of the others.

Examining the Given Polynomials

The specific polynomials under consideration are:

• P1(x) = 2x^2 + 1
• P2(x) = -4x^2 + x
• P3(x) = x - 2

Our goal is to determine whether these three polynomials are linearly dependent or independent. To do this, we will analyze whether there exist nontrivial scalars \( c1, c2, c_3 \) such that:

\[ c1 (2x^2 + 1) + c2 (-4x^2 + x) + c_3 (x - 2) = 0 \]

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Methodology for Testing Linear Dependence

Step 1: Write the Linear Combination Explicitly

Express the combination as:

\[ c1 (2x^2 + 1) + c2 (-4x^2 + x) + c_3 (x - 2) = 0 \]

which expands to:

\[ (2c1 x^2 + c1) + (-4c2 x^2 + c2 x) + (c3 x - 2c3) = 0 \]

Step 2: Group Like Terms

Combine coefficients for similar powers of \( x \):

\[ (2c1 - 4c2) x^2 + (c2 + c3) x + (c1 - 2c3) = 0 \]

Since this must hold for all \( x \), the coefficients of each power must individually equal zero:

\[
\begin{cases}
2c1 - 4c2 = 0 \quad \text{(coefficient of } x^2 \text{)} \\
c2 + c3 = 0 \quad \text{(coefficient of } x \text{)} \\
c1 - 2c3 = 0 \quad \text{(constant term)} \\
\end{cases}
\]

Step 3: Solve the System of Equations

Solve for \( c1, c2, c_3 \):

1. From the first equation:

\[ 2c1 = 4c2 \Rightarrow c1 = 2c2 \]

1. From the second:

\[ c3 = -c2 \]

1. From the third:

\[ c1 = 2c3 \] But since \( c1 = 2c2 \) and \( c3 = -c2 \), substitute:

\[ 2c2 = 2(-c2) \Rightarrow 2c2 = -2c2 \Rightarrow 4c2 = 0 \Rightarrow c2 = 0 \]

Then:

• \( c_2 = 0 \)


• \( c3 = -c2 = 0 \)


• \( c1 = 2c2 = 0 \)

All scalars are zero, indicating the only solution is the trivial one.

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Interpretation and Conclusion

Implication of the Result

Since the only solution to the linear combination equaling zero is when all scalars are zero, the set of polynomials \(\{P1, P2, P_3\}\) is linearly independent.

Final Answer

Based on this computation, the statement "The Polynomials P1 = 2x^2+1, P2= - 4x^2+x And P3 =x-2 Are Linearly Dependent" is False.

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Key Points Summary

• Linear dependence involves expressing one polynomial as a combination of others.
• To test dependence, set up a linear combination and equate to the zero polynomial.
• Group coefficients by powers of \( x \) and solve the resulting system.
• The unique trivial solution indicates linear independence.
• For the polynomials provided, the only solution is trivial, confirming independence.

Why Is This Important in Mathematics?

Understanding whether a set of polynomials is linearly dependent or independent is crucial in various mathematical applications, including:

• Determining bases for polynomial vector spaces
• Simplifying polynomial systems
• Analyzing the span of polynomial sets
• Applications in approximation theory, coding theory, and computational algebra

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Additional Resources for Learning Polynomial Linear Dependence

• Linear Algebra textbooks focusing on vector spaces
• Online tutorials on polynomial vector spaces
• Video lectures on linear dependence and independence
• Interactive algebra software to experiment with polynomial combinations

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In Summary

The detailed analysis of the polynomials \( P1, P2, P_3 \) demonstrates that they are linearly independent. This conclusion is reached by solving the linear combination equations and confirming that the only solution is the trivial one where all coefficients are zero. Recognizing such independence is vital in understanding the structure and basis of polynomial spaces, which has broad applications across mathematics and engineering.

Remember: Always verify linear dependence or independence through systematic algebraic methods to ensure accurate understanding of polynomial relationships.

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Meta Description: Discover how to determine whether the polynomials \( P1=2x^2+1 \), \( P2=-4x^2+x \), and \( P_3=x-2 \) are linearly dependent or independent. Learn step-by-step methods and key concepts in linear algebra.