Polynomials Test - 11

The degree of a quadratic polynomial is 2.

The correct answer is B.

x³ + y³ + z³ − 3xyz = (x + y + z) {x² + y² + z² − xy − yz − zx)

= (x + y + z) {x² + y² + z² + 2xy + 2yz + 2zx − 3xy − 3yz − 3zx}

= (x + y + z) {(x + y + z)² − 3(xy + yz + zx)} … (i)

It is given that: (x + y + z)² = 16

∴ x + y + z = ± 4

However, x, y, and z are all positive integers.

∴ x + y + z = 4

Substituting x + y + z = 4 and xy + yz + zx = 5 in equation (i):

x³ + y³ + z³ − 3xyz = (4) {16 − 3(5)} = 4

The correct answer is B.

According to factor theorem, if x − a is a factor of a polynomial f(x), then f(a) = 0

(x − 3) is a factor of x³ + px² − 9.

∴(3)³ + p(3)² − 9 = 0

⇒ 27 + 9p − 9 = 0

⇒ 9p = −18

⇒ p = −2

x − 3 is a factor of 2x² + qx − 39.

∴2(3)² + q(3) − 39 = 0

⇒ 18 + 3q − 39 = 0

⇒ 3q = 21

⇒ q = 7

Thus, the values of p and q are −2 and 7 respectively.

The correct answer is A.

It is known that a polynomial of degree n can have at most n zeroes.

The degree of the polynomial x³ − 2x² − 7x + 12 is 3.

Thus, the given polynomial can have at most three zeroes.

The correct answer is C.