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

To find the polynomial \( p(x) \) given that its zeroes are \(-3\) and \(8\), we need to use the fact that if a polynomial has zeroes \( \alpha \) and \( \beta \), it can be expressed in the form:

\(p(x) = k(x - \alpha)(x - \beta)\)

Here, \( \alpha = -3 \) and \( \beta = 8 \). Therefore, the polynomial can be written as:

\(p(x) = k(x + 3)(x - 8)\)

In the options provided, option B is:

\((x + 3)(-x + 8)\)

This is equivalent to:

\((x + 3)(-1)(x - 8) = - (x + 3)(x - 8)\)

Since the polynomial is not required to have a specific leading coefficient, and option B correctly represents the root structure, it matches the requirement where the zeroes are at \(-3\) and \(8\). Thus, it fits the problem's condition.

Let's verify the factorization:

• First, expand \( (x + 3)(x - 8) \):
• \((x + 3)(x - 8) = x^2 - 8x + 3x - 24 = x^2 - 5x - 24\)
• The expression given in option B, \((x + 3)(-x + 8)\), is simply the negative of this result, which does not affect the roots.

Therefore, the correct polynomial \( p(x) \) that has zeroes at \(-3\) and \(8\) can be represented as \((x + 3)(-x + 8)\).

Hence, the correct answer is:

Option B: \((x + 3)(-x + 8)\)