NCERT Solutions Class 10 Maths Chapter 2 — Polynomials

Solution:

Step 1: By definition, a number 'k' is called a zero of a polynomial P(x) if P(k) = 0.

Step 2: This means that when 'k' is substituted for 'x' in the polynomial, the entire expression evaluates to zero.

Solution:

Step 1: A quadratic polynomial is a polynomial of degree 2. This means the highest power of the variable in the expression must be 2.

Step 2: Option A (2x + 5) is a linear polynomial (degree 1). [Degree of 2x + 5 is 1]

Step 3: Option B (3x² - 4x + 1) has the highest power of x as 2, so it is a quadratic polynomial. [Degree of 3x² - 4x + 1 is 2]

Step 4: Option C (x³ + 2x - 7) is a cubic polynomial (degree 3). [Degree of x³ + 2x - 7 is 3]

Step 5: Option D (4x⁴) is a polynomial of degree 4. [Degree of 4x⁴ is 4]

Solution:

Step 1: The zeros of a polynomial P(x) are the x-values for which P(x) = 0. Geometrically, these are the points where the graph of y = P(x) intersects or touches the x-axis.

Step 2: If the graph intersects the x-axis at exactly three distinct points, it means there are three distinct x-values for which P(x) = 0.

Step 3: Therefore, P(x) has exactly three real zeros.

Solution:

Step 1: For a quadratic polynomial ax² + bx + c, the sum of its zeros (α + β) is given by the formula -b/a.

Step 2: In the given polynomial x² - 7x + 10, we have a = 1, b = -7, and c = 10.

Step 3: Substituting these values into the formula, the sum of zeros = -(-7)/1 = 7/1 = 7.

Step 4: Ravi's statement that the sum of zeros is 7 is correct, and the reasoning for option D correctly explains why.

Solution:

Step 1: For a quadratic polynomial, if α and β are its zeros, then the polynomial can be written in the form x² - (sum of zeros)x + (product of zeros).

Step 2: Given: Sum of zeros (α + β) = -5.

Step 3: Given: Product of zeros (αβ) = 6.

Step 4: Substitute these values into the general form: x² - (-5)x + (6) = x² + 5x + 6.