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Class 10 — Mathematics (NCERT)

Polynomials: Zeroes, Coefficients and Division Algorithm

Class 10

  • ✓By the end of this lesson students will be able to understand the relationship between zeroes and coefficients of quadratic and cubic polynomials.
  • ✓By the end of this lesson students will be able to apply the division algorithm for polynomials.
  • ✓By the end of this lesson students will be able to find zeroes of a polynomial when some of its zeroes are given.
  • ✓By the end of this lesson students will be able to construct a quadratic polynomial given the sum and product of its zeroes.

Key concepts

Zeroes of a Polynomial

A real number 'k' is said to be a zero of a polynomial p(x) if p(k) = 0. Geometrically, the zeroes of a polynomial are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.

Relationship between Zeroes and Coefficients of a Quadratic Polynomial

For a quadratic polynomial p(x) = ax² + bx + c, where a ≠ 0, if α and β are its zeroes, then:\n1. Sum of zeroes: α + β = -b/a\n2. Product of zeroes: αβ = c/a\n\nA quadratic polynomial whose zeroes are α and β can be written as k[x² - (α + β)x + αβ], where k is any non-zero real number.

Sum of zeroes = -b/a\nProduct of zeroes = c/a
Relationship between Zeroes and Coefficients of a Cubic Polynomial

For a cubic polynomial p(x) = ax³ + bx² + cx + d, where a ≠ 0, if α, β and γ are its zeroes, then:\n1. Sum of zeroes: α + β + γ = -b/a\n2. Sum of the products of the zeroes taken two at a time: αβ + βγ + γα = c/a\n3. Product of zeroes: αβγ = -d/a

Sum of zeroes = -b/a\nSum of products of zeroes taken two at a time = c/a\nProduct of zeroes = -d/a
Division Algorithm for Polynomials

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find 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).\nHere, p(x) is the dividend, g(x) is the divisor, q(x) is the quotient, and r(x) is the remainder.

p(x) = g(x) × q(x) + r(x)

Key facts to remember

  • 1For a quadratic polynomial ax² + bx + c, sum of zeroes = -b/a and product of zeroes = c/a.
  • 2For a cubic polynomial ax³ + bx² + cx + d, sum of zeroes = -b/a, sum of products of zeroes taken two at a time = c/a, and product of zeroes = -d/a.
  • 3If α and β are the zeroes of a quadratic polynomial, the polynomial can be written as k[x² - (α + β)x + αβ].
  • 4The Division Algorithm for Polynomials states p(x) = g(x) × q(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x).
  • 5If a polynomial p(x) has a zero 'k', then (x - k) is a factor of p(x).
  • 6If a polynomial has two zeroes, say 'a' and 'b', then (x-a)(x-b) is a factor. Dividing the polynomial by this factor can help find the remaining zeroes.

Worked examples

Example 1

Find the zeroes of the quadratic polynomial x² + 7x + 10, and verify the relationship between the zeroes and the coefficients.

ILet p(x) = x² + 7x + 10.
IITo find the zeroes, we set p(x) = 0:\nx² + 7x + 10 = 0
IIIFactorise the quadratic polynomial by splitting the middle term:\nx² + 5x + 2x + 10 = 0
IVx(x + 5) + 2(x + 5) = 0
V(x + 5)(x + 2) = 0
VIThis gives x + 5 = 0 or x + 2 = 0.
VIISo, x = -5 or x = -2.
VIIIThe zeroes are α = -5 and β = -2.
9Now, we verify the relationship between zeroes and coefficients.
10Comparing x² + 7x + 10 with ax² + bx + c, we have a = 1, b = 7, c = 10.
11Sum of zeroes: α + β = -5 + (-2) = -7.
12From coefficients: -b/a = -(7)/1 = -7.
13L.H.S. = R.H.S. for sum of zeroes.
14Product of zeroes: αβ = (-5) × (-2) = 10.
15From coefficients: c/a = 10/1 = 10.
16L.H.S. = R.H.S. for product of zeroes.

Answer

The zeroes of the polynomial are -5 and -2. The relationship between zeroes and coefficients is verified.

Always check both sum and product relationships for verification.

Example 2

Divide the polynomial p(x) = 3x³ + x² + 2x + 5 by the polynomial g(x) = 1 + 2x + x² and verify the division algorithm.

IFirst, arrange the terms of the divisor g(x) in descending order of their degrees: g(x) = x² + 2x + 1.
IIPerform polynomial long division:
III 3x - 5
IV ________________
Vx²+2x+1 | 3x³ + x² + 2x + 5
VI -(3x³ + 6x² + 3x)
VII ________________
VIII -5x² - x + 5
9 -(-5x² - 10x - 5)
10 ________________
11 9x + 10
12So, the quotient q(x) = 3x - 5 and the remainder r(x) = 9x + 10.
13Now, we verify the division algorithm: p(x) = g(x) × q(x) + r(x).
14R.H.S. = (x² + 2x + 1) × (3x - 5) + (9x + 10)
15 = x²(3x - 5) + 2x(3x - 5) + 1(3x - 5) + 9x + 10
16 = (3x³ - 5x²) + (6x² - 10x) + (3x - 5) + 9x + 10
17 = 3x³ + (-5x² + 6x²) + (-10x + 3x + 9x) + (-5 + 10)
18 = 3x³ + x² + 2x + 5
19This is equal to p(x).
20Also, the degree of r(x) (degree 1) is less than the degree of g(x) (degree 2).

Answer

The quotient q(x) = 3x - 5 and the remainder r(x) = 9x + 10. The division algorithm is verified.

Always arrange polynomials in descending powers of the variable before division.

Example 3

Obtain all other zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its zeroes are √(5/3) and -√(5/3).

ISince √(5/3) and -√(5/3) are two zeroes, (x - √(5/3)) and (x + √(5/3)) are factors of the given polynomial.
IITherefore, their product (x - √(5/3))(x + √(5/3)) = x² - (5/3) is a factor of the polynomial.
IIITo simplify calculations, we can say that 3(x² - 5/3) = 3x² - 5 is also a factor of the polynomial.
IVNow, we divide the given polynomial p(x) = 3x⁴ + 6x³ - 2x² - 10x - 5 by 3x² - 5.
VPerform polynomial long division:
VI x² + 2x + 1
VII _________________
VIII3x²-5 | 3x⁴ + 6x³ - 2x² - 10x - 5
9 -(3x⁴ - 5x²)
10 _________________
11 6x³ + 3x² - 10x
12 -(6x³ - 10x)
13 _________________
14 3x² - 5
15 -(3x² - 5)
16 _________________
17 0
18The quotient is q(x) = x² + 2x + 1.
19The other zeroes will be the zeroes of the quotient q(x).
20Set q(x) = 0: x² + 2x + 1 = 0
21Factorise the quadratic: (x + 1)² = 0
22This gives x + 1 = 0, so x = -1 (repeated root).
23Thus, the other zeroes are -1 and -1.

Answer

The other zeroes of the polynomial are -1 and -1.

If a polynomial has a zero 'a', then (x-a) is a factor. If 'a' and 'b' are zeroes, then (x-a)(x-b) is a factor.

Common mistakes

  • ✗Incorrectly applying the signs in the zeroes-coefficients relationships (e.g., using b/a instead of -b/a for sum of zeroes).
  • ✗Errors in polynomial long division, especially with signs during subtraction.
  • ✗Not arranging the terms of the dividend and divisor in descending powers of the variable before division.
  • ✗Forgetting to find the zeroes of the quotient polynomial when asked to find 'all other zeroes'.
  • ✗Confusing the sum of products of zeroes taken two at a time (c/a) with the product of all zeroes (-d/a) for cubic polynomials.

Exam tips

  • ★Memorise the relationships between zeroes and coefficients for both quadratic and cubic polynomials thoroughly.
  • ★Practice polynomial long division extensively to avoid calculation errors and ensure accuracy.
  • ★Always verify your division by checking if p(x) = g(x) × q(x) + r(x) and if deg r(x) < deg g(x).
  • ★When finding all zeroes, ensure you factorise the quotient completely to find all possible roots.

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