Class 10 / Maths
Class 10 Maths Polynomials Notes
Notes on zeros of polynomials and the relationship between zeros and coefficients.
Topic introduction
Polynomials in Class 10 focus on zeros, graphs, and the relationship between zeros and coefficients. The chapter is scoring when signs are handled carefully and each zero is verified by substitution.
Key points
- A zero of a polynomial p(x) is a value of x for which p(x) = 0.
- For ax^2 + bx + c, sum of zeros = -b/a and product of zeros = c/a.
- A quadratic polynomial can be formed from zeros alpha and beta as k[x^2 - (alpha + beta)x + alpha beta].
- The graph of a quadratic polynomial may cut the x-axis at two points, one point, or no point.
Definitions
Polynomial: An algebraic expression in which variables have non-negative integral powers.
Zero of a polynomial: A value of x for which p(x) becomes zero.
Quadratic polynomial: A polynomial of degree 2, usually written as ax^2 + bx + c.
Formulas
- For ax^2 + bx + c: sum of zeros = -b/a
- For ax^2 + bx + c: product of zeros = c/a
- Polynomial from zeros alpha and beta: k[x^2 - (alpha + beta)x + alpha beta]
Examples
- x^2 - 5x + 6 has zeros 2 and 3.
- For 3x^2 - 8x + 4, sum of zeros = 8/3 and product = 4/3.
- Zeros 4 and 7 form x^2 - 11x + 28, or any non-zero multiple of it.
Zeros of a polynomial
To check whether a number is a zero, substitute it into the polynomial. If the value becomes zero, the number is a zero of the polynomial.
For linear polynomials, there is one zero. For quadratic polynomials, there can be two real zeros, one repeated zero, or no real zero.
Relationship between zeros and coefficients
For a quadratic polynomial ax^2 + bx + c, use sum = -b/a and product = c/a. Write a, b, and c carefully with their signs.
These formulas help find missing coefficients, verify zeros, and form new polynomials.
Common mistakes
- Using b/a instead of -b/a for sum of zeros.
- Forgetting that a non-zero constant multiple gives an equivalent polynomial with the same zeros.
- Not substituting values to verify zeros.
- Changing signs incorrectly while factorising.
- Confusing degree of polynomial with number of terms.
Quick practice
- Find the zeros of x^2 - 5x + 6.
- For 2x^2 - 7x + 3, find the sum and product of zeros.
- Form a quadratic polynomial whose zeros are 3 and -2.
- Check whether x = 1 is a zero of x^2 - 1.
Related practice
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