Anzal Khan

Anzal Khan

Answered question

2022-08-30

Answer & Explanation

karton

karton

Expert2023-06-03Added 613 answers

To determine whether the polynomials u, v, and w in P(t) are linearly dependent or independent, we need to examine if there exist scalars a, b, and c (not all zero) such that the linear combination au+bv+cw equals the zero polynomial.
Let's analyze each polynomial separately:
1. Polynomial u is not provided in the question. Please provide the expression for polynomial u so that I can proceed with the solution.
2. Polynomial v=t24t+3t+3:
Expanding this polynomial, we have v=t2t+3.
If we assume that au+bv+cw=0 (the zero polynomial), then we can express this equation as:
a(t24t+3t+3)+b(t2+2t2+4t1)+c(2t3t2+3t+5)=0.
Simplifying this equation gives us:
(a+b)t2+(7a+6b+3c)t+(3ab+5c+3)=0.
For this equation to hold true for all t, the coefficients of each term must be zero. Therefore, we have the following system of equations:
a+b=0,
7a+6b+3c=0,
3ab+5c+3=0.
Solving this system of equations will help us determine if v is linearly dependent or independent.
3. Polynomial w=2t3t2+3t+5:
Using the same logic as above, if we assume that au+bv+cw=0, we can express this equation as:
a(t24t+3t+3)+b(t2+2t2+4t1)+c(2t3t2+3t+5)=0.
Simplifying this equation gives us:
(b+2c)t2+(7a+6b+3c)t+(3ab+5c+3)=0.
Again, for this equation to hold true for all t, the coefficients of each term must be zero. Thus, we have the following system of equations:
b+2c=0,
7a+6b+3c=0,
3ab+5c+3=0.
Solving this system of equations will determine if w is linearly dependent or independent.

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