Polynomial Zeros Representation

We are given two polynomials, \(p(x)\) and \(q(x)\), both of degree 4 with a leading coefficient of 1:

\( p(x) = x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 \)

\( q(x) = x^4 + b_3x^3 + b_2x^2 + b_1x + b_0 \)

The zeros of \(p(x)\) are given as \(\alpha, \beta, \gamma, \delta\).

The zeros of \(q(x)\) are given as \(\alpha, \beta, \gamma, \lambda\).

Deriving Polynomial Expressions

Using the factor theorem, a polynomial can be expressed as the product of its linear factors corresponding to its zeros. Since the leading coefficient of both \(p(x)\) and \(q(x)\) is 1, we can express them as:

\( p(x) = (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) \)

\( q(x) = (x - \alpha)(x - \beta)(x - \gamma)(x - \lambda) \)

Calculating Polynomial Difference

We need to calculate the difference between the two polynomials, \(p(x) - q(x)\):

\( p(x) - q(x) = [ (x - \alpha)(x - \beta)(x - \gamma)(x - \delta) ] - [ (x - \alpha)(x - \beta)(x - \gamma)(x - \lambda) ] \)

Notice that the term \((x - \alpha)(x - \beta)(x - \gamma)\) is common to both parts. We can factor it out:

\( p(x) - q(x) = (x - \alpha)(x - \beta)(x - \gamma) \left[ (x - \delta) - (x - \lambda) \right] \)

Now, simplify the expression inside the square brackets:

\( p(x) - q(x) = (x - \alpha)(x - \beta)(x - \gamma) \left[ x - \delta - x + \lambda \right] \)

\( p(x) - q(x) = (x - \alpha)(x - \beta)(x - \gamma) (\lambda - \delta) \)

Simplifying the Quotient Expression

The question asks us to find the value of the expression \(\frac{p(x) - q(x)}{(x - \alpha) (x - \beta) (x - \gamma)}\).

Substitute the expression we found for \(p(x) - q(x)\) into the fraction:

\( \frac{p(x) - q(x)}{(x - \alpha) (x - \beta) (x - \gamma)} = \frac{(x - \alpha)(x - \beta)(x - \gamma) (\lambda - \delta)}{(x - \alpha) (x - \beta) (x - \gamma)} \)

Provided that \(x \neq \alpha\), \(x \neq \beta\), and \(x \neq \gamma\), we can cancel the common factor \((x - \alpha)(x - \beta)(x - \gamma)\) from both the numerator and the denominator.

\( \frac{p(x) - q(x)}{(x - \alpha) (x - \beta) (x - \gamma)} = \lambda - \delta \)

Final Simplified Result

Therefore, the expression \(\frac{p(x) - q(x)}{(x - \alpha) (x - \beta) (x - \gamma)}\) simplifies to \(\lambda - \delta\).