{"slug":"cbse-class-10-maths-polynomials","title":"Polynomials","description":"Degrees and types of polynomials, the geometrical meaning of zeroes, and the relationship between zeroes and coefficients, weighted towards finding zeroes and forming a quadratic from its zeroes as set across the 2026 papers.","board":"CBSE","grade":"Class 10","subject":"Maths","chapter":"Polynomials","url":"https://www.flipnlearn.app/deck/cbse-class-10-maths-polynomials","studyUrl":"https://www.flipnlearn.app/deck/cbse-class-10-maths-polynomials/study","cards":[{"id":"degree-of-a-polynomial","kind":"term-definition","front":"Degree of a polynomial","back":"The highest power of the variable in the polynomial. For instance 4x + 2 has degree 1, 2y² − 3y + 4 has degree 2 and 5x³ − 4x² + x − 2 has degree 3; polynomials of degree one, two and three are named linear, quadratic and cubic."},{"id":"zero-of-a-linear-polynomial","kind":"formula","front":"Zero of a linear polynomial ax + b","back":"−b/a, that is −(constant term) ÷ (coefficient of x), where a ≠ 0. For 2x + 3 the zero is −3/2, the point where the line y = 2x + 3 crosses the x-axis."},{"id":"quadratic-polynomial","kind":"term-definition","front":"Quadratic polynomial","back":"A polynomial of degree 2, whose general form in x is ax² + bx + c with a, b and c real and a ≠ 0. The name comes from 'quadrate', meaning square."},{"id":"cubic-polynomial","kind":"term-definition","front":"Cubic polynomial","back":"A polynomial of degree 3, whose most general form is ax³ + bx² + cx + d with a, b, c and d real and a ≠ 0. Examples include x³, 2 − x³ and 3x³ − 2x² + x − 1."},{"id":"zero-of-a-polynomial","kind":"term-definition","front":"Zero of a polynomial","back":"A real number k for which p(k) = 0. For p(x) = x² − 3x − 4, both p(−1) and p(4) equal 0, so −1 and 4 are its zeroes."},{"id":"geometrical-meaning-of-zeroes","kind":"term-definition","front":"Geometrical meaning of zeroes","back":"The zeroes of p(x) are exactly where the graph of y = p(x) crosses or touches the x-axis: each zero is the x-coordinate of such a point. So −1 and 4, the zeroes of x² − 3x − 4, are where its parabola cuts the x-axis."},{"id":"how-many-zeroes-a-quadratic-can-have","kind":"concept-question","front":"In how many ways can the graph of a quadratic polynomial meet the x-axis, and what does each mean for its zeroes?","back":"The graph is a parabola, opening upwards if a > 0 and downwards if a < 0. It can cut the x-axis at two distinct points, giving two zeroes; touch it at one point, giving two equal zeroes; or miss it entirely, giving no zero."},{"id":"maximum-number-of-zeroes","kind":"formula","front":"Maximum number of zeroes","back":"A polynomial of degree n has at most n zeroes, because its graph meets the x-axis in at most n points. So a quadratic has at most 2 zeroes and a cubic at most 3."},{"id":"how-to-count-zeroes-from-a-graph","kind":"worked-step","front":"How do you find the number of zeroes of p(x) from the graph of y = p(x)?","back":"Count the points where the graph meets the x-axis; the x-coordinate of each is a zero. A graph meeting the x-axis at one point gives one zero, at two points two zeroes, and so on."},{"id":"sum-of-zeroes-of-a-quadratic","kind":"formula","front":"Sum of zeroes of a quadratic","back":"α + β = −b/a = −(coefficient of x) ÷ (coefficient of x²), for zeroes α and β of ax² + bx + c. For 2x² − 8x + 6, whose zeroes are 1 and 3, the sum 4 equals −(−8)/2."},{"id":"product-of-zeroes-of-a-quadratic","kind":"formula","front":"Product of zeroes of a quadratic","back":"αβ = c/a = (constant term) ÷ (coefficient of x²), for zeroes α and β of ax² + bx + c. For 2x² − 8x + 6, whose zeroes are 1 and 3, the product 3 equals 6/2."},{"id":"why-sum-and-product-of-zeroes-follow-from-factors","kind":"concept-question","front":"Why are the sum and product of the zeroes of ax² + bx + c equal to −b/a and c/a?","back":"If α and β are zeroes, then (x − α) and (x − β) are factors, so ax² + bx + c = k[x² − (α + β)x + αβ]. Comparing coefficients gives a = k, b = −k(α + β) and c = kαβ, which rearrange to α + β = −b/a and αβ = c/a."},{"id":"quadratic-polynomial-from-its-zeroes","kind":"formula","front":"Quadratic polynomial with given zeroes α and β","back":"k[x² − (α + β)x + αβ] for a non-zero real constant k. In words: x² − (sum of zeroes)x + (product of zeroes), multiplied by any such constant."},{"id":"how-to-find-zeroes-and-verify-coefficients","kind":"worked-step","front":"How do you find the zeroes of a quadratic by factorising and verify the zeroes-coefficients relationship?","back":"Factorise, set each factor equal to zero, then compare the sum and product of the zeroes with −b/a and c/a. In the chapter's example, x² + 7x + 10 = (x + 2)(x + 5) gives zeroes −2 and −5, whose sum −7 equals −7/1 and whose product 10 equals 10/1."},{"id":"how-to-form-a-quadratic-from-sum-and-product","kind":"worked-step","front":"How do you find a quadratic polynomial when the sum and product of its zeroes are given?","back":"Take a = 1, so that b = −(sum) and c = product, and write x² − (sum)x + product. In the chapter's example, a sum of −3 and a product of 2 give x² + 3x + 2; every other answer has the form k(x² + 3x + 2)."},{"id":"relations-between-zeroes-and-coefficients-of-a-cubic","kind":"formula","front":"Relationships between zeroes and coefficients of a cubic polynomial","back":"If α, β and γ are the zeroes of ax³ + bx² + cx + d, then α + β + γ = −b/a, αβ + βγ + γα = c/a and αβγ = −d/a."}]}