We are tasked with determining if the first polynomial can be expressed as a linear combination of the other two. For the given polynomials in $P_{\leq 3}(\mathbb{R})$, the question is to check if the first polynomial $p_1(x) = x^3 + 3x^2 - 1$ can be expressed as a linear combination of the second polynomial $p_2(x) = x^3 - 3x + 5$ and the third polynomial $p_3(x) = x^3 + 2x^2 - x + 1$.

This can be done by checking if there exist scalars $a$ and $b$ such that:

$p_1(x) = a \cdot p_2(x) + b \cdot p_3(x)$

Substitute the given polynomials into this expression:

$x^3 + 3x^2 - 1 = a(x^3 - 3x + 5) + b(x^3 + 2x^2 - x + 1)$

Now, expand the right-hand side:

$a(x^3 - 3x + 5) = a \cdot x^3 - 3a \cdot x + 5a$ $b(x^3 + 2x^2 - x + 1) = b \cdot x^3 + 2b \cdot x^2 - b \cdot x + b$

Combining the two expansions, we get:

$a \cdot x^3 - 3a \cdot x + 5a + b \cdot x^3 + 2b \cdot x^2 - b \cdot x + b = (a + b) \cdot x^3 + 2b \cdot x^2 + (-3a - b) \cdot x + (5a + b)$

Now, equate the coefficients of the powers of $x$ from both sides:

1. Coefficient of $x^3$: $a + b = 1$
2. Coefficient of $x^2$: $2b = 3$ From this, we get $b = \frac{3}{2}$.
3. Coefficient of $x$: $-3a - b = 0$ Substituting $b = \frac{3}{2}$, we get: $-3a - \frac{3}{2} = 0 \quad \Rightarrow \quad a = -\frac{1}{2}$
4. Constant term: $5a + b = -1$ Substituting $a = -\frac{1}{2}$ and $b = \frac{3}{2}$, we get: $5\left(-\frac{1}{2}\right) + \frac{3}{2} = -\frac{5}{2} + \frac{3}{2} = -1$ This is true.

Since all the equations are satisfied, the first polynomial $p_1(x) = x^3 + 3x^2 - 1$ can be expressed as a linear combination of the other two polynomials.

The linear combination is:

$p_1(x) = -\frac{1}{2} \cdot p_2(x) + \frac{3}{2} \cdot p_3(x)$

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Would you like further details or have any questions?

Here are 5 related questions you might find helpful:

1. Can any polynomial in $P_{\leq 3}(\mathbb{R})$ always be expressed as a linear combination of others in the same space?
2. What does it mean for polynomials to be linearly dependent in a vector space?
3. How can the concept of basis polynomials be applied to higher-degree polynomials?
4. How would the solution change if we had an extra polynomial to consider?
5. What is the geometric interpretation of expressing one polynomial as a linear combination of others?

Tip: To solve systems of equations like this efficiently, matrix methods such as Gaussian elimination can be used.