Final answer:
Without knowing the specific value of coefficient 'b', we cannot definitively determine the number of real roots of the polynomial equation x⁴−x³+bx²−x+1. The polynomial could have multiple real roots, but exact analysis requires more information. The correct answer is (c) The equation has no real roots.
Step-by-step explanation:
The question is asking about the number of real roots for the polynomial equation x⁴−x³+bx²−x+1. Knowing that a polynomial of degree n has exactly n roots (real or complex) according to the Fundamental Theorem of Algebra, and not all roots are necessarily real, we need to analyze the equation further.
For a polynomial to have all real roots, it must change signs as many times as there are roots when evaluated across different x values in the real number line. Let's observe the equation: By analyzing the coefficients and the constant term, we can see that there are changes in sign, which could mean roots are present.
However, using Descartes' Rule of Signs or plotting the graph requires more specific information about the coefficient b. Therefore, without additional information, we cannot definitively determine the exact number of real roots. We need b to apply the quadratic formula or any other method to find the roots or the nature of the roots.
In this equation, the coefficients are all positive (or zero), so we can conclude that there are no sign changes. Therefore, the equation has no real roots.