To determine the value of the coefficient [tex]\( c \)[/tex] in a polynomial given the sum and product of its zeroes, we need to recall a few key concepts about quadratic polynomials.
For a quadratic polynomial of the form [tex]\( ax^2 + bx + c \)[/tex], the following relationships hold true:
1. The sum of the roots (zeroes) is given by [tex]\( -\frac{b}{a} \)[/tex].
2. The product of the roots is given by [tex]\( \frac{c}{a} \)[/tex].
Given:
- The sum of the roots is [tex]\( \frac{2}{5} \)[/tex].
- The product of the roots is [tex]\( \frac{1}{4} \)[/tex].
We aim to determine the value of the coefficient [tex]\( c \)[/tex] based on these relationships. Let's proceed step-by-step:
1. Sum of the roots: This implies [tex]\( -\frac{b}{a} = \frac{2}{5} \)[/tex].
2. Product of the roots: This implies [tex]\( \frac{c}{a} = \frac{1}{4} \)[/tex].
To simplify our calculations, we assume the leading coefficient [tex]\( a = 1 \)[/tex]. This is common practice to make the algebra simpler without loss of generality, as the relationships are directly proportional.
Given [tex]\( a = 1 \)[/tex]:
- The product of the roots equation becomes [tex]\( c = \frac{1}{4} \)[/tex].
Therefore, the value of the coefficient [tex]\( c \)[/tex] is [tex]\( \frac{1}{4} \)[/tex], which numerically is [tex]\( 0.25 \)[/tex].
Since [tex]\( 0.25 \)[/tex] is not listed in the options directly, let’s reinterpret the numerical result. The closest interpretation from the given choices is the one that matches our calculation directly.
None of the given options (1, 5, 2, 4) precisely match [tex]\( 0.25 \)[/tex]. This suggests a check or a potential issue with the problem setup or the choices offered. Based on the calculations above, our determined coefficient is indeed 0.25.
However, if forced to choose from the given options, we should reconsider the problem setup or seek further clarification, but our precise value remains [tex]\( 0.25 \)[/tex] as calculated above.
