Polynomials and Algebraic Identities (Class 9)

What this worksheet covers

This is an eighteen-question practice worksheet on polynomials and algebraic identities for Class 9, written to the pattern of the SOF International Mathematics Olympiad (IMO). It carries 24 marks and is designed to be attempted in 45 minutes, split across three sections — Mathematical Reasoning, Everyday Mathematics and Reasoning and Achievers Section. Every question is multiple choice with exactly one correct option, and every one comes with a fully worked solution rather than just an answer key — so a child who gets one wrong can see where the reasoning went astray instead of only learning that it did.

Skills your child practises

• Applied problem-solving on polynomials and algebraic identities at Class 9 level
• Working accurately against a time limit, as the real paper demands
• Mathematical Reasoning questions in the olympiad format
• Everyday Mathematics and Reasoning questions in the olympiad format
• Achievers Section questions in the olympiad format
• Reading worked solutions to find the step that went wrong

How to use it

1. 1There are 18 questions in this worksheet. All questions are compulsory.
2. 2Each question has four options (A), (B), (C) and (D). Exactly one option is correct.
3. 3Questions 1–15 carry 1 mark each. Questions 16–18 (Achievers Section) carry 3 marks each.
4. 4There is no negative marking. Use of a calculator is not allowed.
5. 5Rough work may be done on the blank space at the side of each page.

Sample questions from this worksheet

A preview of what's inside Polynomials and Algebraic Identities (Class 9) — the PDF has the full set, and every question comes with a worked solution like these.

1. 1. The degree of the polynomial 5x³ − 4x² + 7x − 2 is _______.

   • A1
   • B2
   • C3correct
   • D5
   Solution: The degree is the highest power of the variable that appears with a non-zero coefficient. Here the powers are 3, 2, 1 and 0, so the degree is 3. (5 is the leading coefficient, not the degree.)

2. 2. If (x − 2) is a factor of x³ − 3x² + kx − 4, then the value of k is _______.

   • A2
   • B−4
   • C4correct
   • D6
   Solution: By the Factor Theorem, (x − 2) is a factor exactly when p(2) = 0. So 2³ − 3(2²) + 2k − 4 = 8 − 12 + 2k − 4 = 2k − 8 = 0, giving k = 4.

3. 3. The polynomial p(x) = x³ + ax² + bx + 6 has (x − 2) as a factor and leaves a remainder 3 when divided by (x − 3). The values of a and b are _______.

   • Aa = 3, b = −1
   • Ba = −1, b = −3
   • Ca = −3, b = 1
   • Da = −3, b = −1correct
   Solution: Factor Theorem: p(2) = 8 + 4a + 2b + 6 = 0, so 4a + 2b = −14, that is 2a + b = −7. Remainder Theorem: p(3) = 27 + 9a + 3b + 6 = 3, so 9a + 3b = −30, that is 3a + b = −10. Subtracting the first relation from the second gives a = −3, and then b = −7 − 2(−3) = −1. So a = −3 and b = −1.