A polynomial is an algebraic expression consisting of variables and coefficients combined using addition subtraction and multiplication. It has terms with non-negative integer exponents. The degree is the highest power of the variable. Polynomials are used in various fields like physics engineering and economics to model real-world situations and solve problems efficiently.
Class 10 Maths Chapter 2 Polynomials focuses on understanding zeros and coefficients of polynomials along with division algorithms. It covers linear quadratic and cubic polynomials. This chapter prepares students for solving real-life problems and strengthens algebraic skills essential for CBSE Exam 2024-25 and higher studies in mathematics ensuring a clear grasp of concepts before advancing to complex topics.
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Step 1: Polynomial Structure
Given polynomial: f(x) = x³ + 3x² + 3x + 1
Step 2: Divisibility Test Method
– To check if a polynomial is divisible by (x + a), we can use the remainder theorem
– The polynomial is divisible by (x + a) if f(-a) = 0
Step 3: Checking Possible Divisors
Let’s evaluate f(-1):
f(-1) = (-1)³ + 3(-1)² + 3(-1) + 1
= -1 + 3 – 3 + 1
= 0
Mathematical Insight:
– When f(-1) = 0, it means (x + 1) is a factor of the polynomial
– This indicates the polynomial is completely divisible by (x + 1)
Step 4: Verification
Dividing x³ + 3x² + 3x + 1 by (x + 1):
– The result is a quadratic: x² + 2x + 1
– (x + 1)(x² + 2x + 1) = x³ + 3x² + 3x + 1 ✓
Conclusion:
The polynomial is divisible by x + 1.
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