Question

Assertion (A) : The polynomial \(p(y) = y^{2 + 4y + 3}\) has two zeroes.
Reason (R) : A quadratic polynomial can have at most two zeroes.

• A. Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

For a quadratic polynomial \(ax^{2} + bx + c\), calculate the discriminant \(D = b^{2} - 4ac\).
If \(D>0\), there are 2 distinct zeroes.
If \(D = 0\), there is 1 repeated zero.
If \(D<0\), there are no real zeroes.
In all cases, the count is \(\le 2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The degree of a polynomial determines the maximum number of zeroes it can have. A quadratic polynomial is of degree 2.
Step 2: Detailed Explanation:
Evaluating Assertion (A):
The given polynomial is \(p(y) = y^{2} + 4y + 3\).
To find the zeroes, solve \(y^{2} + 4y + 3 = 0\):
\[ y^{2} + 3y + y + 3 = 0 \]
\[ y(y + 3) + 1(y + 3) = 0 \]
\[ (y + 1)(y + 3) = 0 \]
Zeroes are \(y = -1\) and \(y = -3\).
The polynomial has exactly two zeroes. Thus, Assertion (A) is true.
Evaluating Reason (R):
By the fundamental theorem of algebra, a polynomial of degree \(n\) has at most \(n\) real zeroes. For a quadratic polynomial (\(n=2\)), it can have 0, 1, or 2 zeroes. Thus, "at most two zeroes" is a true mathematical statement. Reason (R) is true.
Relating (A) and (R):
Since \(p(y)\) is a quadratic polynomial, it follows the rule stated in Reason (R). The fact that a quadratic polynomial can have at most two zeroes is the reason why this specific quadratic polynomial has two zeroes (and not more).
Step 3: Final Answer:
Both (A) and (R) are true and (R) is the correct explanation of (A).