To tackle the problem of factoring the polynomial [tex]\(x^2 - x + 7\)[/tex], let's examine it in detail.
Consider the quadratic polynomial [tex]\(x^2 - x + 7\)[/tex]. To factor this polynomial, we typically look for two binomials [tex]\((x - a)(x - b)\)[/tex] whose product equals the given polynomial. To do this, the coefficients and constants will need to match the original polynomial once expanded.
1. Identify Possible Factor Pairs: Let's assume it can be factored into [tex]\((x - a)(x - b)\)[/tex].
2. Expand the Binomials: By expanding [tex]\((x - a)(x - b)\)[/tex], we get:
[tex]\[
(x - a)(x - b) = x^2 - (a + b)x + ab
\][/tex]
In this case, we need:
- The sum of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to equal -1 (since the coefficient of [tex]\(x\)[/tex] is -1).
- The product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] to equal 7, the constant term.
3. Solve the System of Equations:
- We need [tex]\(a + b = 1\)[/tex]
- We need [tex]\(ab = 7\)[/tex]
Let's find pairs [tex]\((a, b)\)[/tex] that satisfy these conditions adequately:
- The pairs [tex]\((a, b)\)[/tex] that multiply to 7 could be [tex]\((1, 7)\)[/tex], [tex]\((-1, -7)\)[/tex], [tex]\((\sqrt{7}, -\sqrt{7})\)[/tex], etc.
- None of these pairs will simultaneously sum up to -1.
4. Conclusion:
After analyzing the polynomial, it becomes apparent that there are no pairs of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] consisting of real numbers that satisfy both conditions [tex]\(a + b = 1\)[/tex] and [tex]\(ab = 7\)[/tex]. As a result:
Therefore, the polynomial [tex]\(x^2 - x + 7\)[/tex] cannot be factored into simpler binomials with real coefficients. Hence, the correct answer is:
- Cannot be factored
