Question

If the zeroes of a polynomial \(p(x)\) are \(-3\) and \(8\), then \(p(x)\) equals

• A. \(x^2 + 5x - 4\)
• B. \((x + 3)(-x + 8)\)
• C. \(a(x^2 + 5x - 24)\)
• D. \(x^2 - 24\)

Hint:

Check roots by substitution! Plug \(x = -3\) and \(x = 8\) into the options. The one that results in zero for both values is the correct polynomial.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A polynomial with zeroes \(\alpha\) and \(\beta\) can be written in factored form as \(p(x) = k(x - \alpha)(x - \beta)\), where \(k\) is a non-zero constant.
Step 2: Detailed Explanation:
Given zeroes: \(\alpha = -3\) and \(\beta = 8\).
The factored form is:
\[ p(x) = k(x - (-3))(x - 8) \]
\[ p(x) = k(x + 3)(x - 8) \]
Let's check the options:
(A) \(x^2 + 5x - 4\): Sum of roots is \(-5\). Incorrect.
(B) \((x + 3)(-x + 8)\): Here, if we set the expression to \(0\), we get \(x + 3 = 0 \implies x = -3\) and \(-x + 8 = 0 \implies x = 8\). This matches the given zeroes perfectly. Note that this is just the form with \(k = -1\).
(C) \(a(x^2 + 5x - 24)\): For this, sum of roots is \(-5\). Incorrect.
(D) \(x^2 - 24\): Roots are \(\pm \sqrt{24}\). Incorrect.
Step 3: Final Answer:
The polynomial is \((x + 3)(-x + 8)\).