Find the span of the following set of polynomials:


x^2 − x + 1, 2x + 1

To find the span of the given set of polynomials \(\{x^2 - x + 1, 2x + 1\}\), we need to determine the set of all linear combinations of these two polynomials.

### Step 1: General Linear Combination

Let \( p(x) = a(x^2 - x + 1) + b(2x + 1) \), where \(a\) and \(b\) are constants. We want to express the general form of this linear combination.

\[
p(x) = a(x^2 - x + 1) + b(2x + 1)
\]
Expanding this expression:

\[
p(x) = a(x^2) + a(-x) + a(1) + b(2x) + b(1)
\]
\[
p(x) = a x^2 + (-a + 2b)x + (a + b)
\]

Thus, the general polynomial \(p(x)\) formed by a linear combination of \(x^2 - x + 1\) and \(2x + 1\) is:

\[
p(x) = a x^2 + (-a + 2b)x + (a + b)
\]

### Step 2: Span

The span of the set of polynomials \(\{x^2 - x + 1, 2x + 1\}\) consists of all polynomials of the form \( a x^2 + (-a + 2b)x + (a + b) \), where \(a\) and \(b\) are real numbers.

Thus, the span is the set of all polynomials of degree at most 2, with a free \(a\) for the quadratic term, and the coefficients of \(x\) and the constant term being dependent on both \(a\) and \(b\). Therefore, the span is:

\[
\text{Span}\{x^2 - x + 1, 2x + 1\} = \{a x^2 + (-a + 2b)x + (a + b) : a, b \in \mathbb{R}\}
\]

### Conclusion:

The span is the set of all polynomials of degree at most 2 where the coefficients of \(x^2\), \(x\), and the constant term can be adjusted by choosing different values of \(a\) and \(b\).

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Would you like to see this expanded into more detail? Here are five questions you could explore next:

1. How do we find the dimension of the span of these polynomials?
2. What would happen if we added a third polynomial, such as \(x^3\), to the set?
3. Can you provide specific examples of polynomials that are in the span of this set?
4. How can we find a basis for the span of these polynomials?
5. What role do linear independence and dependence play in determining the span?

**Tip:** When determining the span of vectors (or polynomials), it's useful to explore the linear independence of the given set to understand its dimensionality better.

Find the span of the following set of polynomials: x^2 − x + 1, 2x + 1

Discover how to determine the span of the set of polynomials {x^2 − x + 1, 2x + 1} through linear combinations. This problem explores key concepts in linear algebra and polynomial functions, involving linear combinations and the span of a set in vector spaces. Suitable for advanced high school or undergraduate students.

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