Certainly! Let's go through the detailed, step-by-step solution to find the value of [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} + 1 \)[/tex] when [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] are the zeroes (roots) of the polynomial [tex]\( 5x^2 + 3x - 7 \)[/tex].
### Step 1: Finding the Sum and Product of the Zeros
For a polynomial given by [tex]\( ax^2 + bx + c \)[/tex], the sum and product of its roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] can be found using Vieta's formulas:
- Sum of the roots ([tex]\( \alpha + \beta \)[/tex]):
[tex]\[
\alpha + \beta = -\frac{b}{a}
\][/tex]
- Product of the roots ([tex]\( \alpha \beta \)[/tex]):
[tex]\[
\alpha \beta = \frac{c}{a}
\][/tex]
For the polynomial [tex]\( 5x^2 + 3x - 7 \)[/tex]:
- Here, [tex]\( a = 5 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -7 \)[/tex].
- Therefore,
[tex]\[
\alpha + \beta = -\frac{b}{a} = -\frac{3}{5} = -0.6
\][/tex]
[tex]\[
\alpha \beta = \frac{c}{a} = \frac{-7}{5} = -1.4
\][/tex]
### Step 2: Expressing [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} \)[/tex] in Terms of the Sum and Product of the Roots
We need to evaluate [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} \)[/tex]. Using the identity for reciprocals of the roots:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}
\][/tex]
Substituting the values we found:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-0.6}{-1.4} = 0.4285714285714286
\][/tex]
### Step 3: Adding 1 to the Result
We need to find [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} + 1 \)[/tex]:
[tex]\[
\frac{1}{\alpha} + \frac{1}{\beta} + 1 = 0.4285714285714286 + 1 = 1.4285714285714286
\][/tex]
Therefore, the value of [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} + 1 \)[/tex] is:
[tex]\[
1.4285714285714286
\][/tex]
So, detailing the answer, we have:
- The sum of the zeros [tex]\( (\alpha + \beta) \)[/tex] is [tex]\(-0.6\)[/tex].
- The product of the zeros [tex]\( (\alpha \beta) \)[/tex] is [tex]\(-1.4\)[/tex].
- The reciprocal sum [tex]\( \left(\frac{1}{\alpha} + \frac{1}{\beta}\right) \)[/tex] is [tex]\(0.4285714285714286\)[/tex].
- Finally, the value of [tex]\( \frac{1}{\alpha} + \frac{1}{\beta} + 1 \)[/tex] is [tex]\(1.4285714285714286\)[/tex].
