State whether the following statements are true or false:

(1) is a zero of the linear polynomial p(x) = 5x + 7.

(2) p(x) = x² + 2x + 1 has two distinct zeros.

(3) The cubic polynomial p(x) = x³ + x² — x — 1 has two distinct zeros.

(4) The graph of the cubic polynomial p(x) = x³ meets the X—axis at only one point.

(5) Any quadratic polynomial p(x) has at least one zero, x R

(1) p(x) = 5x + 7

∴ p = 5 + 7 = 14

⇒ p ≠0

⇒ is not a zero of p(x).

So the statement is False.

(2) Given, p(x) = x² + 2x + 1

To find the zeros of p(x), let p(x) = 0

∴ x² + 2x + 1 = 0

∴ (x + 1)² = 0 Using the identity: (a+ b) ² = (a² + 2ab + b²)

∴ x + 1 = 0 or x = –1

∴ x = –1 or x = –1

Here, both the zeros are equal, i.e. –1, and hence not distinct.

So the statement is False.

(3) Given, p(x) = x³ + x² — x — 1

To find the zeros of p(x), let p(x) = 0

∴ x³ + x² — x — 1 = 0

∴ x² (x + 1) – 1 (x + 1) = 0

∴ (x² – 1)(x + 1) = 0

∴ (x – 1) (x + 1)(x + 1) = 0 Using the identity: (a² – b²) = (a – b) (a + b)

∴ x – 1 = 0 or x + 1 = 0 or x + 1 = 0

∴ x = 1 or x = –1 or x = –1

∴ Two distinct zeros of p(x) are 1 and –1.

Hence, p(x) has at the most two distinct zeros.

So the statement is True.

(4) Given, p(x) = x³

To find the zeros of p(x), let p(x) = 0

∴x³ = 0

⇒ x = 0.

⇒ The graph of p(x) = x³ meets the X-axis at only one point i.e. (0, 0).

So the statement is True.

(5) If the graph of the quadratic polynomial p(x) does not intersect the x-axis at any point, then the quadratic polynomial does not have any real zero.

So the statement is False.