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. For example 3x² + 2x – 5 is a polynomial. 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 solid foundation in polynomial concepts and their applications.
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Finding the Second Zero of a Quadratic Polynomial
Step 1: Understanding the Given Information
– Polynomial: x² – 7x + 10
– One known zero: 5
Step 2: Verification of the Known Zero
Let’s first verify that 5 is indeed a zero:
5² – 7(5) + 10 = 25 – 35 + 10 = 0
Step 3: Using Vieta’s Formulas
In a quadratic polynomial ax² + bx + c, if p and q are zeros:
– Sum of zeros: p + q = -b/a
– Product of zeros: p * q = c/a
For x² – 7x + 10:
– a = 1
– b = -7
– c = 10
Step 4: Finding the Second Zero
We are aware that one zero is 5, therefore let’s use the variable x to represent the second zero.
Sum of zeros formula:
5 + x = 7
x = 7 – 5
x = 2
Verification:
– First zero: 5
– Second zero: 2
– Check sum: 5 + 2 = 7
– Check product: 5 * 2 = 10
Mathematical Insight:
Vieta’s formulas offer a beautiful method of determining polynomial zeros
without resorting to complicated solving methods. They show the profound
connection between a polynomial’s coefficients and its roots.
Conclusion:
The other zero of the polynomial is 2.
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