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:

If you are unsure of the algebraic expansion, simply substitute the given zeroes into the options.
For option (B), at $x = -3$, $(x+3)(-x+8) = 0$, and at $x = 8$, $(x+3)(-x+8) = 0$.
Since both values satisfy the polynomial, it must be the correct option!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given that the zeroes of a quadratic polynomial $p(x)$ are $\alpha = -3$ and $\beta = 8$.
We need to identify which of the given expressions correctly represents a polynomial with these zeroes.

Step 2: Key Formula or Approach:
A quadratic polynomial with zeroes $\alpha$ and $\beta$ can be written in its factored form as:
\[ p(x) = k(x - \alpha)(x - \beta) \]
where $k$ is any non-zero real constant.
In expanded form, this is:
\[ p(x) = k\left[x^2 - (\alpha + \beta)x + \alpha\beta\right] \]

Step 3: Detailed Explanation:

• Let us first find the sum and product of the zeroes:
\[ \text{Sum of zeroes } (\alpha + \beta) = -3 + 8 = 5 \]
\[ \text{Product of zeroes } (\alpha\beta) = (-3) \times 8 = -24 \]

• Write the standard polynomial form:
\[ p(x) = k(x^2 - 5x - 24) \]

• Let us analyze the options:
- Option (A): $x^2 + 5x - 4$ does not match.
- Option (C): $a(x^2 + 5x - 24)$ has a positive linear term coefficient, which is incorrect since the sum is $+5$, making the term $-5x$.
- Option (D): $x^2 - 24$ is missing the linear term.
- Option (B): Let us expand $(x + 3)(-x + 8)$:
\[ (x + 3)(-x + 8) = -x^2 + 8x - 3x + 24 = -x^2 + 5x + 24 \]
If we factor out $-1$:
\[ -x^2 + 5x + 24 = -(x^2 - 5x - 24) \]
Comparing this with $k(x^2 - 5x - 24)$, we see that this is a valid quadratic polynomial where $k = -1$.
Therefore, the polynomial in option (B) has the correct roots.


Step 4: Final Answer:
The polynomial $p(x)$ is equal to $(x + 3)(-x + 8)$.