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 \(\lambda - \delta\).
Similar Questions
- Let \(p(x)\) be a polynomial. When \(p(x)\) is divided by \((x-1)\), it leaves 2 as the remainder. When \(p(x)\) is divided by \((x-2)\), it leaves 1 as the remainder. What is the remainder when \(p(x)\) is divided by \((x - 1)(x-2)\)?
- What is the remainder when \(x^6\) is divided by \(x^2 + 1\)?
- \((x+2)\) is a factor of which one of the following?
What is the HCF of the polynomials x⁸ + x⁴ + 1 and x⁴ + x² + 1?
- If 2 is a zero of the polynomial \(p(x) = x^3 + 3x^2 - 6x - a\), then what is the sum of the squares of the other zeros of the polynomial?
- Consider the following in respect of the polynomial \(x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6}\):
1. The remainder is zero when the polynomial is divided by \(x^2 + 1\).
2. The remainder is zero when the polynomial is divided by \(x^4 + 1\).
Which of the statements given above is/are correct? - If \(x^3 + px^2 + qx + r\) is an integer for all integral values of \(x\), then consider the following statements :
I. \(p\) must be an integer
II. \(q\) must be an integer
III. \(r\) must be an integer
Which of the statements given above is/are correct? - Which of the following expressions can divide both the polynomials \(x^3+2x^2-5x+2\) and \(x^3+4x^2+x-6\) exactly ?
I. \(x-1\)
II. \(x+1\)
III. \(x+2\)
Select the correct answer using the code given below : If x² - 5x + 4 is a factor of x⁴ - px² + q, then what are the values of p and q respectively?
Important Questions from Polynomials
If y 2= y + 7, then what is the value of y 3?
Factorize x 2- y 2- 9z 2+ 6yz
If one of the zeros of the polynomial x 3+ ax 2+ bx + c is - 1, then the product of other two zeros is equal to :
If a(a + b + c) 2 = 1792; b(a + b + c) 2 = 1536; c(a + b + c) 2 = 768, then what will be the value of b?
If x = 3 so, what is the value of x 2 + 2x + 5 ?
