Polynomials Solved Examples (Class 9 Mathematics)
Factor polynomials and apply the remainder and factor theorems. These problems develop polynomial manipulation skills and algebraic reasoning.
TL;DR: Factor polynomials and apply the remainder and factor theorems. These problems develop polynomial manipulation skills and algebraic reasoning.
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Factor polynomials and apply the remainder and factor theorems. These problems develop polynomial manipulation skills and algebraic reasoning.
Polynomials — Solved Numerical Examples (Step by Step)
Example 1: Divide the polynomial p(x) = x³ - 3x² + 4x - 12 by (x - 3) using the Remainder Theorem.
Solution: By Remainder Theorem, the remainder when p(x) is divided by (x - 3) is p(3)
p(3) = 3³ - 3(3)² + 4(3) - 12 = 27 - 27 + 12 - 12 = 0
So remainder = 0, meaning (x - 3) is a factor
Example 2: Find the remainder when p(x) = 2x³ + 3x² - 5x + 7 is divided by (x + 2).
Solution: By Remainder Theorem, remainder = p(-2)
p(-2) = 2(-2)³ + 3(-2)² - 5(-2) + 7
p(-2) = 2(-8) + 3(4) + 10 + 7 = -16 + 12 + 10 + 7 = 13
Example 3: Factorize: x² + 7x + 10
Solution: Find two numbers that multiply to 10 and add to 7: these are 5 and 2
x² + 7x + 10 = (x + 5)(x + 2)
Example 4: Factorize: 2x² - 5x - 3
Solution: Find factors of 2 × (-3) = -6 that add to -5: these are -6 and 1
2x² - 5x - 3 = 2x² - 6x + x - 3 = 2x(x - 3) + 1(x - 3) = (2x + 1)(x - 3)
Example 5: Check if (x - 2) is a factor of p(x) = x³ - 4x² + 4x - 8.
Solution: By Factor Theorem, (x - 2) is a factor if p(2) = 0
p(2) = 2³ - 4(2)² + 4(2) - 8 = 8 - 16 + 8 - 8 = -8
Since p(2) ≠ 0, (x - 2) is NOT a factor
Example 6: Factorize: x² - 16
Solution: This is a difference of squares: a² - b² = (a + b)(a - b)
x² - 16 = x² - 4² = (x + 4)(x - 4)
Tips
- Remainder Theorem: When p(x) is divided by (x - a), the remainder is p(a).
- Factor Theorem: (x - a) is a factor of p(x) if and only if p(a) = 0.
- For ax² + bx + c factorization, find two numbers that multiply to ac and add to b.
Frequently Asked Questions
How do you know if a polynomial is completely factored?
A polynomial is completely factored when all factors are either linear (degree 1) or irreducible quadratics (cannot be factored further over the real numbers).
What is the difference between a root and a factor?
If r is a root of p(x), then p(r) = 0. The corresponding factor is (x - r). Roots and factors are related but not the same concept.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
- Pair of Linear Equations in Two Variables
- Circles
- Statistics
- Linear Equations in Two Variables
- Heron's Formula
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