Courses
By Shailendra Singh
|
Updated on 11 Nov 2025, 18:34 IST
A polynomial is an algebraic expression made up of variables and coefficients, combined using addition, subtraction, and multiplication, where powers of variables are non-negative integers. For example, 2u3 + 3u2 - 2u + 4 is a polynomial in the variable u.
The expression aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ is called a polynomial, where aₙ, aₙ₋₁, … are coefficients and x is the variable. Terms like 1/x, √x, or x⁻² are not polynomials because they do not meet the required powers or structure.
5xx^2 - 7x^3 + 2x - 1The degree of a polynomial is the highest power of the variable in the expression. For p(x) = 3x^4 + 2x^2 - x + 5, the degree is 4.
| Type | Description | Example |
| Monomial | One term | 7x2 |
| Binomial | Two terms | x2 + 5x |
| Trinomial | Three terms | 3x3 - x + 7 |
Based on degree:
x + 3x^2 + 2x + 1x^3 - 4x + 5A polynomial function maps values from domain to range by plugging in real numbers into a polynomial equation like f(x) = 2x^2 + 3x - 1. When substituting values for x, you get the output called the "value of the polynomial at that point".
Zeros (roots) of a polynomial are values of x for which p(x) = 0. Geometrically, zeros correspond to the points where the graph intersects the x-axis. For a quadratic (ax^2 + bx + c), there can be up to 2 zeros; for cubic, up to 3 zeros.
Loading PDF...
Combine like terms (same power of variable):
(3x2 + 2x) + (4x2 - x) = 7x2 + x
(5x2 + x) - (2x2 + 3x) = 3x2 - 2x
Multiply each term of the first polynomial by every term of the second:
(x + 2)(x - 3) = x2 - x - 6
Long division is used to divide one polynomial (dividend) by another (divisor), resulting in quotient and remainder polynomials. If remainder is zero, the divisor is a factor. Follows the process similar to regular number division.
Factoring means writing a polynomial as a product of its factors.
Quick Example: x2 - 5x + 6 = (x - 2)(x - 3)
Calculators help quickly evaluate polynomials, perform operations, and check zeros. Enter coefficients as per calculator type and use polynomials mode if available.
JEE
NEET
Foundation JEE
Foundation NEET
CBSE
| Name | Formula | Explanation |
| General Polynomial | p(x) = aₙ xⁿ + ... + a₁x + a₀ | Standard form |
| Degree | Highest power of variable | E.g., in 2x⁴ + x², degree = 4 |
| Value at x = a | p(a) | Substitute a and calculate |
| Zero | p(r) = 0 | Value of x that makes p(x) = 0 |
| Linear Polynomial | ax + b | 1 zero, straight line graph |
| Quadratic Formula | x = [-b ± √(b²-4ac)]/(2a) | Zeros of quadratic polynomials |
| Sum of Zeros (Quad) | a + b = -b/a | For ax² + bx + c |
| Product of Zeros | ab = c/a | Quadratic polynomial |
| Polynomial Division | p(x) = g(x) × q(x) + r(x) | Division Algorithm: Dividend = Divisor × Quotient + Remainder |
Polynomials are basic to algebra, graphing, calculus, and competitive exam success. Every step in operations, factoring, and division is logical, rooted in mathematical principles, and applicable across sciences. Concepts, formulas, and problem-solving techniques presented above are based on CBSE syllabus and classroom standards.
These Class 10 CBSE notes on polynomials cover definitions, types, functions, operations, degree, zeros, and practical steps for calculation and factoring. Trust the step-by-step methods and formula table above for strong exam performance and clear algebra concepts.
No courses found
A polynomial is an algebraic expression that combines one or more terms, each consisting of a variable raised to a non-negative integer power, multiplied by a coefficient. For example, 2u3 + 3u2 + 2u + 4 is a polynomial in variable u. The powers must be positive whole numbers, and coefficients are real numbers. Polynomials do not have variables in denominators, negative exponents, or within roots.
Polynomials can be classified by the number of terms (monomial: 1 term, binomial: 2 terms, trinomial: 3 terms), by their degree (linear, quadratic, cubic, quartic), or by the number of variables (univariate, bivariate, multivariate). For example, a linear polynomial has degree 1 (like x + 2), while a quadratic has degree 2 (like x2 + 3x + 1).
The degree of a polynomial is the highest exponent of the variable in the expression. For instance, for p(x) = 4x2 + 3x + 5, the degree is 2 because 2 is the largest power of x present. The zero polynomial (all coefficients zero) has an undefined degree.
Zeros (or roots) of a polynomial are real numbers for which the polynomial equals zero. For example, in p(x) = x2 - 4, the zeros are x = 2 and x = -2 because replacing x with these values makes the polynomial zero. Zeros are crucial for solving equations and understanding the graph of a polynomial.
To find the value of a polynomial at a certain number, substitute the value into every instance of the variable in the expression. For example, for p(x) = x3 - 6x2 + 11x - 6, to find p(2), calculate 23 - 6(2)2 + 11(2) - 6, which gives 0.
For a quadratic polynomial ax2 + bx + c, if α and β are its zeros, then:
This relationship is often used to construct or analyze quadratic equations.
The division algorithm states that any polynomial f(x) can be divided by a non-zero polynomial g(x), resulting in a quotient q(x) and a remainder r(x), so f(x) = g(x)q(x) + r(x), where the degree of r(x) is less than the degree of g(x). If the remainder is zero, g(x) is a factor of f(x).
Polynomials are used in various real-life situations, such as predicting growth, modeling curves, financial calculations, and physics problems. For example, the trajectory of a thrown ball or the shape of a roller coaster can be represented by polynomial equations.
Yes, polynomials can include more than one variable (like x, y, z). These are called multivariate polynomials, and they're used in problems where multiple factors affect results, such as area calculations (length × width).
The zeros of a polynomial are the points where its graph intersects the x-axis. For a quadratic polynomial, the graph is a parabola, and its zeros are the points where the curve crosses the x-axis. A cubic polynomial can cross the axis once, twice, or three times, showing up to three zeros.