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

To solve this problem, let's analyze both the assertion (A) and the reason (R) provided.

• Assertion (A): The polynomial \( p(y) = y^2 + 4y + 3 \) has two zeroes.
  • A quadratic polynomial is in the form \( ax^2 + bx + c \), where \( a \neq 0 \). In this case, \( a = 1 \), \( b = 4 \), and \( c = 3 \).
  • By the Fundamental Theorem of Algebra, a quadratic polynomial can have at most two zeroes.
  • The zeroes of the polynomial can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Substituting the values, we have \( a = 1 \), \( b = 4 \), and \( c = 3 \): \(x = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times 3}}{2 \times 1} = \frac{-4 \pm \sqrt{16 - 12}}{2} = \frac{-4 \pm \sqrt{4}}{2} = \frac{-4 \pm 2}{2}\).
  • This results in two solutions: \( x = -1 \) and \( x = -3 \), indicating that the polynomial has exactly two zeroes.
• Reason (R): A quadratic polynomial can have at most two zeroes.
  • The statement correctly identifies the nature of a quadratic polynomial, which, due to its degree being 2, can have at most two solutions/zeroes.

Both the assertion and the reason are true, and the reason serves as a correct explanation for the assertion since it explicitly states that the polynomial \( p(y) = y^2 + 4y + 3 \) is quadratic and therefore can have two zeroes.

Correct Answer: Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).