Class 10 Mathematics · Chapter 2 NotesPolynomials

A polynomial in one variable is an expression in which the variable appears only with whole-number powers, and the highest such power is called the degree. A polynomial of degree 1 is linear, of degree 2 is quadratic, and of degree 3 is cubic. The general quadratic polynomial is ax² + bx + c with a ≠ 0, and the general cubic polynomial is ax³ + bx² + cx + d with a ≠ 0. Expressions that contain the variable in a denominator or under a square root, such as 1/(x − 1), are not polynomials. The value of a polynomial p(x) at x = k is written p(k) and is found by substituting k for x.

• Degreehighest power of the variable in the polynomial
• Lineardegree 1, general form ax + b, a ≠ 0
• Quadraticdegree 2, general form ax² + bx + c, a ≠ 0
• Cubicdegree 3, general form ax³ + bx² + cx + d, a ≠ 0

A real number k is called a zero of a polynomial p(x) if p(k) = 0. For a linear polynomial ax + b, the only zero is −b/a, which is the x-coordinate of the point where the line y = ax + b crosses the x-axis. For a quadratic polynomial, the graph y = ax² + bx + c is a parabola that opens upwards when a > 0 and downwards when a < 0. It may cut the x-axis at two distinct points, touch it at exactly one point, or not meet it at all, so a quadratic polynomial can have two distinct zeroes, two equal zeroes, or no real zero. A cubic polynomial can have at most three zeroes, and in general a polynomial of degree n has at most n zeroes.

• Zero of p(x)a real number k with p(k) = 0
• Zero of ax + b is −b/a
• Quadratic polynomialat most 2 zeroes
• Cubic polynomialat most 3 zeroes
• Degree n polynomialat most n zeroes

If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then x − α and x − β are its factors, so ax² + bx + c = k(x − α)(x − β). Expanding and comparing coefficients gives a = k, b = −k(α + β) and c = kαβ. Dividing by a gives the two standard results: the sum of the zeroes equals −b/a, and the product of the zeroes equals c/a. For example, the zeroes of x² + 7x + 10 are −2 and −5; their sum is −7 = −(7)/1 and their product is 10 = 10/1, matching the coefficients. These formulas let you check your factorisation and also build a polynomial from its zeroes.

• Sum of zeroesα + β = −b/a = −(coefficient of x)/(coefficient of x²)
• Product of zeroesαβ = c/a = (constant term)/(coefficient of x²)
• If sum = S and product = P, a quadratic polynomial is x² − Sx + P (or any non-zero multiple of it)

A cubic polynomial ax³ + bx² + cx + d has three zeroes, say α, β and γ. Comparing the expanded form k(x − α)(x − β)(x − γ) with the polynomial gives three relationships. The sum of the zeroes is −b/a. The sum of the products of the zeroes taken two at a time, αβ + βγ + γα, is c/a. The product of all three zeroes, αβγ, is −d/a. For instance, for 2x³ − 5x² − 14x + 8 the zeroes are 4, −2 and 1/2; their sum is 5/2 = −(−5)/2, the sum of pairwise products is −7 = −14/2, and the product is −4 = −8/2. These identities are useful for verifying zeroes without fully factorising.

• α + β + γ = −b/a
• αβ + βγ + γα = c/a
• αβγ = −d/a

If you are told the sum S and the product P of the zeroes of a quadratic polynomial, you can write one such polynomial as x² − Sx + P. For example, if the sum is −3 and the product is 2, then S = −3 and P = 2, so the polynomial is x² − (−3)x + 2 = x² + 3x + 2. Any non-zero multiple k(x² + 3x + 2) also has the same zeroes, so the answer is not unique unless a leading coefficient is specified. This method works because expanding (x − α)(x − β) gives x² − (α + β)x + αβ, which directly connects the coefficients to the sum and product of the zeroes.

• Given sum S and product P, use x² − Sx + P
• Multiply by any non-zero constant k to get another valid polynomial
• Check by expanding (x − α)(x − β)

Just as you divide whole numbers, you can divide one polynomial by another. If p(x) and g(x) are polynomials with g(x) ≠ 0, and the degree of g(x) is less than or equal to the degree of p(x), then there exist unique polynomials q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree of g(x). Here p(x) is the dividend, g(x) the divisor, q(x) the quotient and r(x) the remainder. This is the division algorithm for polynomials. It is used to check whether one polynomial is a factor of another and to find unknown coefficients when a remainder is given.

• Dividend = Divisor × Quotient + Remainder
• Remainder is zero or has degree less than the divisor
• If r(x) = 0, then g(x) is a factor of p(x)

Factorization is the main tool for finding zeroes by hand. For a quadratic polynomial, split the middle term into two terms whose coefficients multiply to give the product of the leading coefficient and the constant term, then group and factor. For example, 2x² − 8x + 6 becomes 2x² − 6x − 2x + 6 = 2x(x − 3) − 2(x − 3) = 2(x − 1)(x − 3), so the zeroes are 1 and 3. Identities such as a² − b² = (a − b)(a + b) help with special cases like x² − 3 = (x − √3)(x + √3). For cubic polynomials, once one zero is known, dividing by the corresponding linear factor reduces the problem to a quadratic that can be factorised further.

• Split the middle term to factorise quadratics
• Use a² − b² = (a − b)(a + b) for difference of squares
• For cubics, find one zero and divide to get a quadratic factor