Polynomials — Chapter Test | Class 9 IMO

Q1

If x + y + z = 0, then the value of x²/yz + y²/zx + z²/xy is:

Taking the common denominator xyz gives (x³ + y³ + z³)/xyz. Since x + y + z = 0, x³ + y³ + z³ = 3xyz, so the value is 3xyz/xyz = 3.

Q2

Statement A: A binomial can have degree 100. Statement B: A polynomial of degree 1 can have 3 terms. Which is true?

A is true (e.g. x¹⁰⁰ + 1 is a binomial of degree 100). B is false: a degree-1 polynomial has at most 2 terms (ax + b).

Q3

If x + 1/x = 4, then the value of x⁴ + 1/x⁴ is:

x + 1/x = 4 → x² + 1/x² = 16 − 2 = 14. Squaring again: x⁴ + 1/x⁴ = 14² − 2 = 196 − 2 = 194.

Q4

If x² − 1 is a factor of ax⁴ + bx³ + cx² + dx + e, which relation holds?

x = 1 and x = −1 are roots. p(1) = a + b + c + d + e = 0 and p(−1) = a − b + c − d + e = 0; adding/subtracting gives a + c + e = b + d.

Q5

For a circular ring, the area depends on (R² − r²). If R = 101 m and r = 99 m, find R² − r² without squaring.

R² − r² = (R − r)(R + r) = (101 − 99)(101 + 99) = 2 × 200 = 400.

Q6

A metal sheet of dimensions (x + 5) by (x − 5) has area:

(x + 5)(x − 5) = x² − 25 by the identity (a + b)(a − b) = a² − b².

Q7

What is the degree of the zero polynomial?

A non-zero constant polynomial has degree 0, but the degree of the zero polynomial is not defined.

Q8

Using Pascal's Triangle, what is the sum of the coefficients in the expansion of (x + y)⁵?

Put x = 1 and y = 1: (1 + 1)⁵ = 2⁵ = 32.

Q9

The factorisation of x² − 5x − 6 is:

Split the middle term: x² − 6x + x − 6 = x(x − 6) + 1(x − 6) = (x − 6)(x + 1).

Q10

If both (x + 1) and (x − 1) are factors of ax³ + x² − 2x + b, find a and b.

p(−1) = −a + 1 + 2 + b = 0 → −a + b = −3; p(1) = a + 1 − 2 + b = 0 → a + b = 1. Solving gives b = −1 and a = 2.

Q11

Match each polynomial to its type — (P) 3x² + 5x, (Q) 7x³, (R) 5x + 2 — with (i) Quadratic Binomial, (ii) Cubic Monomial, (iii) Linear Binomial:

3x² + 5x is a quadratic binomial, 7x³ is a cubic monomial, and 5x + 2 is a linear binomial.

Q12

If a² + b² + c² = 250 and ab + bc + ca = 3, find |a + b + c|.

(a + b + c)² = 250 + 2(3) = 256, so |a + b + c| = √256 = 16.

Q13

To find 249² − 248² quickly, which identity helps, and what is the value?

Using a² − b² = (a − b)(a + b): 249² − 248² = (249 − 248)(249 + 248) = 1 × 497 = 497.

Q14

In the sequence P₁(x) = x − 1, P₂(x) = x² − 1, P₃(x) = x³ − 1, …, the common linear factor of every term is:

For any xⁿ − 1, substituting x = 1 gives 0, so (x − 1) is a factor of every term.

Q15

If p(x) = x + 4, then p(x) + p(−x) is equal to:

p(x) = x + 4 and p(−x) = −x + 4, so their sum is (x + 4) + (−x + 4) = 8.