CBSE Class 10 Maths Chapter 2: Polynomials Revision Notes

Key Concepts to Remember

• A polynomial P(x) is an expression of the form a₀ + a₁x + a₂x² + ... + aₙxⁿ, where aᵢ are real numbers and n is a non-negative integer.
• The degree of a polynomial is the highest power of the variable in the polynomial.
• A zero of a polynomial P(x) is a value 'k' such that P(k) = 0. Graphically, zeros are the x-intercepts.
• A linear polynomial (degree 1) has at most one zero. A quadratic polynomial (degree 2) has at most two zeros. A cubic polynomial (degree 3) has at most three zeros.
• For a quadratic polynomial ax² + bx + c, sum of zeros (α+β) = -b/a and product of zeros (αβ) = c/a.
• For a cubic polynomial ax³ + bx² + cx + d, sum of zeros (α+β+γ) = -b/a, sum of products of zeros taken two at a time (αβ+βγ+γα) = c/a, and product of zeros (αβγ) = -d/a.
• The Division Algorithm for Polynomials states: Given polynomials P(x) and G(x) (where G(x) ≠ 0), there exist unique polynomials Q(x) and R(x) such that P(x) = G(x) ⋅ Q(x) + R(x), where R(x) = 0 or degree of R(x) < degree of G(x).

Essential Definitions

Polynomial

An algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

Degree of a Polynomial

The highest power of the variable in a polynomial. For example, in 3x³ + 2x - 5, the degree is 3.

Zero of a Polynomial

A real number 'k' for which the value of the polynomial P(x) becomes zero, i.e., P(k) = 0. Graphically, these are the x-coordinates where the graph intersects the x-axis.

Linear Polynomial

A polynomial of degree 1. Its general form is ax + b, where a ≠ 0.

Quadratic Polynomial

A polynomial of degree 2. Its general form is ax² + bx + c, where a ≠ 0.

Cubic Polynomial

A polynomial of degree 3. Its general form is ax³ + bx² + cx + d, where a ≠ 0.

Monomial

A polynomial with only one term (e.g., 5x²).

Binomial

A polynomial with two terms (e.g., 3x + 7).

Trinomial

A polynomial with three terms (e.g., x² - 2x + 1).

Understanding Polynomials and Their Zeros

A polynomial is a special type of algebraic expression. For an expression to be a polynomial, the powers of the variables must be whole numbers (0, 1, 2, 3, ...), and the coefficients must be real numbers. For instance, 4x^2 - 2x + 7 is a polynomial, but 3x^(1/2) + 5 or 1/x + 2 are not, because x^(1/2) has a fractional power and 1/x implies x^(-1) (negative power). The degree of a polynomial is determined by the highest power of the variable present. For example, 5x^3 + 2x^2 - x + 10 has a degree of 3.

The zeros of a polynomial are the specific values of the variable for which the entire polynomial evaluates to zero. If P(x) is a polynomial, then k is a zero if P(k) = 0. Geometrically, for a polynomial y = P(x), the zeros correspond to the x-coordinates where the graph of y = P(x) intersects the x-axis. A linear polynomial (degree 1) can have at most one zero, meaning its graph (a straight line) intersects the x-axis at most once. A quadratic polynomial (degree 2) has a graph that is a parabola, which can intersect the x-axis at most twice, hence it has at most two zeros. Similarly, a cubic polynomial (degree 3) can have at most three zeros. The number of zeros is always less than or equal to the degree of the polynomial. Understanding zeros is crucial as they provide insights into the behavior and roots of polynomial equations.

Types of Polynomials and Their Zeros

Relationship Between Zeros and Coefficients

• 1. Quadratic Polynomial (ax² + bx + c): If α and β are the zeros, then: Sum of zeros (α + β) = -b/a Product of zeros (αβ) = c/a Example: For x² - 5x + 6, a=1, b=-5, c=6. α + β = -(-5)/1 = 5. (Zeros are 2, 3 so 2+3=5) αβ = 6/1 = 6. (Zeros are 2, 3 so 2*3=6)
• 2. Cubic Polynomial (ax³ + bx² + cx + d): If α, β, and γ are the zeros, then: Sum of zeros (α + β + γ) = -b/a Sum of products of zeros taken two at a time (αβ + βγ + γα) = c/a Product of zeros (αβγ) = -d/a Example: For x³ - 6x² + 11x - 6, a=1, b=-6, c=11, d=-6. α + β + γ = -(-6)/1 = 6. αβ + βγ + γα = 11/1 = 11. αβγ = -(-6)/1 = 6. (Zeros are 1, 2, 3)

Division Algorithm for Polynomials

1. — Arrange the terms of both the dividend P(x) and the divisor G(x) in decreasing order of their degrees. If any power is missing, you can write it with a zero coefficient (e.g., x³ + 0x² + 5x - 2).
2. — Divide the first term of the dividend by the first term of the divisor. This gives the first term of the quotient Q(x).
3. — Multiply the first term of the quotient by the entire divisor G(x) and subtract the result from the dividend. This forms a new dividend.
4. — Bring down the next term(s) of the original dividend to form the new dividend. Repeat Steps 2 and 3 with the new dividend.
5. — Continue the process until the degree of the remainder R(x) is less than the degree of the divisor G(x), or the remainder is 0.

Exam Tip: Common Pitfalls & Scoring Cues

Always double-check signs when applying the formulas for the sum and product of zeros (especially -b/a and -d/a). For polynomial division, be meticulous with subtraction; sign errors are very common. When asked to verify the relationship between zeros and coefficients, ensure you explicitly state the values of a, b, c (and d for cubic) and then substitute them into the formulas. For questions involving finding zeros given some zeros, use the division algorithm effectively. Remember that if (x-k) is a factor, then k is a zero of the polynomial. This helps in factoring higher-degree polynomials.

Quick Revision Check

• Q1: What is the degree of the polynomial 5x⁴ - 3x⁷ + 2x - 9? A1: The degree is 7, as it is the highest power of x.
• Q2: If α and β are the zeros of x² - 7x + 10, find α + β and αβ. A2: For x² - 7x + 10, a=1, b=-7, c=10. α + β = -(-7)/1 = 7. αβ = 10/1 = 10.
• Q3: How many zeros can a cubic polynomial have at most? A3: A cubic polynomial can have at most three zeros.
• Q4: State the Division Algorithm for Polynomials. A4: For any polynomials P(x) and G(x) (where G(x) ≠ 0), there exist unique polynomials Q(x) and R(x) such that P(x) = G(x) ⋅ Q(x) + R(x), where R(x) = 0 or degree of R(x) < degree of G(x).

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