Question

If the sum and product of the zeroes of a quadratic polynomial are 3 and –10 respectively, then the polynomial is

• A. \(x^2-3x-10\)
• B. \(x^2+3x-10\)
• C. \(x^2+3x+10\)
• D. \(x^2-3x+10\)

Answer 1

The Correct Option is A

We are given that the sum and product of the zeroes of a quadratic polynomial are:

\( \alpha + \beta = 3 \), 

\( \alpha \beta = -10 \).

Step 1: General Form of Quadratic Polynomial

The quadratic polynomial whose roots are \( \alpha \) and \( \beta \) is given by:

\[ x^2 - (\alpha + \beta)x + \alpha \beta \]

Step 2: Substituting Given Values

\[ x^2 - (3)x + (-10) \]

\[ x^2 - 3x - 10 \]

Final Answer: \( x^2 - 3x - 10 \).

Answer 2

To find the polynomial given the sum and product of its zeroes, we leverage the relationships provided by Viète's formulas. For a quadratic polynomial of the form \(ax^2 + bx + c\), if \(\alpha\) and \(\beta\) are the zeroes, then:

The sum of the zeroes, \(\alpha + \beta\), is given by \(-b/a\).

The product of the zeroes, \(\alpha \cdot \beta\), is \(c/a\).

In this problem, the sum of the zeroes is 3, and the product is -10. Assuming a monic polynomial (\(a=1\)), the formulas become:

\(\alpha + \beta = -b = 3\) implies \(b = -3\).

\(\alpha \cdot \beta = c = -10\) implies \(c = -10\).

Thus, the polynomial is:

\(x^2 - 3x - 10\)

Therefore, the correct polynomial is \(x^2 - 3x - 10\)