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CBSE · Class 10 · Mathematics

Polynomials 2.3: Relationship between Zeroes and Coefficients

6 concepts · 48 practice questions with answers · Free to read

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Sum and product of the zeroes of a quadratic

From daily life

The two roots of a quadratic carry a secret: their sum and product can be read straight from the coefficients.

Simple definition

For ax² + bx + c with zeroes α and β: α + β = −b/a and αβ = c/a.

Formula: α + β = −b/a; αβ = c/a

ax² + bx + csum = −b/aproduct = c/a

In the figure: For x² − 5x + 6 the zeroes are 2 and 3: sum 5 = −(−5)/1 and product 6 = 6/1.

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Sum and product of the zeroes of a quadratic

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Rule 1: Read from coefficients — Identify a, b, c with their signs.

  • Easy: x² − 5x + 6: sum 5, product 6.
  • Medium: 2x² + 4x − 6: sum −2, product −3.
  • Exam-type: 3x² − x − 4: sum 1/3, product −4/3.

Rule 2: Verify with the zeroes — Add and multiply the actual zeroes to confirm.

  • Easy: x² + x − 6: zeroes 2, −3: sum −1, product −6 ✓.
  • Medium: x² − 4: zeroes 2, −2: sum 0 = −0/1 ✓.
  • Exam-type: x² − 2x + 1: zeroes 1, 1: sum 2, product 1 ✓.
Remember

The sum of the zeroes has the OPPOSITE sign of b/a, the product keeps the sign of c/a.

Common mistake

Forgetting the minus sign in −b/a.

Memory clue

"Sum is minus b over a, product is c over a."

Practice questions

Easy
  1. 1The zeroes of a quadratic polynomial are −6 and −1. What is the product of the zeroes?
  2. 2The zeroes of a quadratic polynomial are −2 and −1. What is the product of the zeroes?
  3. 3Find the product of the zeroes of the polynomial x² + 2x − 8.
Medium
  1. 4Find the sum of the zeroes of the polynomial x² − 9x − 6.
  2. 5The zeroes of a quadratic polynomial are −3 and −6. What is the sum of the zeroes?
  3. 6Find the product of the zeroes of the polynomial x² + 7x + 1.
Challenging
  1. 7Find the sum of the zeroes of the polynomial x² − 6x + 7.
  2. 8What is the sum of the zeroes of 2x² + 9x + 8?
Show answers (try them yourself first)
  1. 16
  2. 22
  3. 3−8
  4. 49
  5. 5−9
  6. 61
  7. 76
  8. 8−9/2

Forming a quadratic from its zeroes

From daily life

Given two roots, we can build a quadratic that has exactly those roots, like building a lock for two keys.

Simple definition

A quadratic with sum S and product P of zeroes is k[x² − Sx + P], for any non-zero k.

Formula: p(x) = k[x² − (α+β)x + αβ]

x² − Sx + PS = α + βP = αβ

In the figure: Zeroes 2 and 3 give S = 5, P = 6, so x² − 5x + 6.

Rule 1: From sum and product — Plug S and P into x² − Sx + P.

  • Easy: S = 3, P = −10: x² − 3x − 10.
  • Medium: S = 0, P = −9: x² − 9.
  • Exam-type: S = −1, P = −2: x² + x − 2.

Rule 2: Fractions as zeroes — Multiply by k to clear the fractions.

  • Easy: Zeroes 1/2, −1: x² + x/2 − 1/2 → 2x² + x − 1.
  • Medium: Zeroes 1/4, −1: 4x² + 3x − 1.
  • Exam-type: Check: 2(1/2)² + 1/2 − 1 = 0 ✓.
Remember

Many polynomials share the same zeroes: multiply by any constant k.

Common mistake

Writing x² + Sx + P instead of x² − Sx + P.

Memory clue

"Minus sum x plus product."

Practice questions

Easy
  1. 1Which quadratic polynomial has zeroes −2 and −8?
  2. 2Which quadratic polynomial has zeroes 2 and 6?
  3. 3Which quadratic polynomial has zeroes −5 and 4?
Medium
  1. 4Which quadratic polynomial has zeroes 5 and 2?
  2. 5Which quadratic polynomial has zeroes 6 and −6?
  3. 6A quadratic polynomial has the sum of its zeroes equal to −2 and the product equal to −6. Which polynomial is it?
Challenging
  1. 7Find the quadratic polynomial (with integer coefficients, leading coefficient 2) whose zeroes are 2 and 1/2.
  2. 8Which quadratic polynomial with integer coefficients has the sum of zeroes −8/3 and the product of zeroes −9/4?
Show answers (try them yourself first)
  1. 1x² + 10x + 16
  2. 2x² − 8x + 12
  3. 3x² + x − 20
  4. 4x² − 7x + 10
  5. 5x² − 36
  6. 6x² + 2x − 6
  7. 72x² − 5x + 2
  8. 812x² + 32x − 27

Zeroes of a quadratic by factorising

From daily life

To find where a thrown ball lands, split the formula into two brackets and set each bracket to zero.

Simple definition

Split the middle term to factorise ax² + bx + c = a(x − α)(x − β). The zeroes are where a bracket is zero. Then check with sum −b/a and product c/a.

x² − 7x + 12(x − 3)(x − 4)zeroes 3, 4

In the figure: x² − 7x + 12 = (x − 3)(x − 4), so the zeroes are 3 and 4.

Rule 1: Factorise and read — Find two numbers with product c and sum b.

  • Easy: x² + x − 6 = (x + 3)(x − 2): zeroes −3, 2.
  • Medium: x² − 9 = (x − 3)(x + 3): zeroes ±3.
  • Exam-type: 2x² − x − 1 = (2x + 1)(x − 1): zeroes −1/2, 1.

Rule 2: Check with the relations — Sum and product must match the coefficients.

  • Easy: 2x² − x − 1: sum 1/2 = −(−1)/2 ✓, product −1/2 ✓.
  • Medium: x² − 7x + 12: sum 7, product 12 ✓.
  • Exam-type: x² + x − 6: sum −1, product −6 ✓.
Remember

If a quadratic does not factorise over whole numbers, its zeroes may still exist (use the formula).

Common mistake

Writing the factor with the wrong sign: the bracket (x + 3) gives the zero −3.

Memory clue

"Bracket zero gives the zero."

Practice questions

Easy
  1. 1Find the zeroes of x² − 196.
  2. 2Find the zeroes of x² − 169.
  3. 3Find the zeroes of x² − 21.
Medium
  1. 4Find the zeroes of x² − 31.
  2. 5The zeroes of x² − 7x + 6 are 1 and 6. What does c/a equal for this polynomial?
  3. 6The zeroes of x² − 12x + 35 are 5 and 7. What does c/a equal for this polynomial?
Challenging
  1. 7Find the zeroes of the quadratic polynomial x² − 3x − 18.
  2. 8Find the zeroes of the quadratic polynomial 10x² − 9x − 9.
Show answers (try them yourself first)
  1. 114 and −14
  2. 213 and −13
  3. 3√21 and −√21
  4. 4√31 and −√31
  5. 56
  6. 635
  7. 7−3 and 6
  8. 83/2 and −3/5

Zeroes and coefficients of a cubic

From daily life

A cubic has three zeroes, and three tidy relations link them to its four coefficients.

Simple definition

For ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a.

Formula: Σα = −b/a; Σαβ = c/a; αβγ = −d/a

x³ − 6x² + 11x − 6sum 6, pairs 11product 6

In the figure: x³ − 6x² + 11x − 6 has zeroes 1, 2, 3: sum 6, pair-sum 11, product 6.

Rule 1: Find the three sums — Read a, b, c, d with signs.

  • Easy: x³ − 6x² + 11x − 6: 6, 11, 6.
  • Medium: x³ − 2x² − x + 2: 2, −1, −2.
  • Exam-type: 2x³ − 4x² + 6x − 8: 2, 3, 4.

Rule 2: Verify with zeroes — Multiply and add the actual zeroes.

  • Easy: Zeroes 1, −1, 2: sum 2 ✓, pairs −1 ✓, product −2 ✓.
  • Medium: x³ − 6x² + 11x − 6: 1×2×3 = 6 = −(−6)/1 ✓.
  • Exam-type: Note the minus sign in −d/a.
Remember

The sign pattern: −b/a, +c/a, −d/a.

Common mistake

Using +d/a for the product of the three zeroes.

Memory clue

"Minus, plus, minus: b, c, d."

Practice questions

Easy
  1. 1For the cubic polynomial ax³ + bx² + cx + d, the sum of the zeroes is ___.
  2. 2A cubic polynomial has zeroes 4, 2 and −4. Find the product of the zeroes.
  3. 3For the cubic polynomial 5x³ + 7x² + 3x − 5, find the sum of the zeroes.
Medium
  1. 4A cubic polynomial has zeroes −5, −7 and 5. Find the sum of the products of zeroes taken two at a time.
  2. 5A cubic polynomial has zeroes −3, 6 and −6. Find the product of the zeroes.
  3. 6Two zeroes of the cubic polynomial x³ + 7x² + 7x − 15 are −5 and 1. Find the third zero.
Challenging
  1. 7Two zeroes of the cubic polynomial x³ + 11x² + 38x + 40 are −2 and −5. Find the third zero.
  2. 8Two zeroes of the cubic polynomial x³ − 12x² + 41x − 30 are 5 and 1. Find the third zero.
Show answers (try them yourself first)
  1. 1−b/a
  2. 2−32
  3. 3−7/5
  4. 4−25
  5. 5108
  6. 6−3
  7. 7−4
  8. 86

Unknown coefficient and the other zero

From daily life

If you know one root, you can find a missing number in the formula, and then the other root.

Simple definition

Put the known zero in p(x) = 0 to find the unknown, or use sum = −b/a and product = c/a to find the other zero.

x² + 3x + k, zero 11 + 3 + k = 0k = −4

In the figure: If 1 is a zero of x² + 3x + k then 1 + 3 + k = 0, so k = −4.

Rule 1: Find the unknown — Substitute the zero, or use sum and product.

  • Easy: x² + 3x + k, zero 1: k = −4.
  • Medium: 2x² − kx + 6, sum 3: k/2 = 3, k = 6.
  • Exam-type: x² − 5x + k, one zero twice the other: α, 2α: 3α = 5, k = 2α² = 50/9.

Rule 2: Find the other zero — Subtract from the sum, or divide the product.

  • Easy: x² − 7x + 10, one zero 2: other = 7 − 2 = 5.
  • Medium: Check by product: 2 × 5 = 10 ✓.
  • Exam-type: x³ − 6x² + 11x − 6, zeroes 1, 2: third = 6 − 3 = 3.
Remember

Both methods should agree: always cross-check with the product.

Common mistake

Putting the zero in the polynomial but forgetting to equate to 0.

Memory clue

"Plug the zero, set it to zero."

Practice questions

Easy
  1. 1One zero of the polynomial x² − 7x + 12 is 4. Find the other zero.
  2. 2One zero of the polynomial x² − 36 is −6. Find the other zero.
  3. 3One zero of the polynomial x² − 11x + 24 is 3. Find the other zero.
Medium
  1. 4For the polynomial x² + 7x + (k + 8), the product of its zeroes is 3. Find k.
  2. 5For the polynomial x² − (k + 1)x + 2, the sum of its zeroes is 7. Find k.
  3. 6One zero of the polynomial x² − 2x − 15 is 5. Find the other zero.
Challenging
  1. 7If the zeroes of 6x² − 2x + (k + 6) are reciprocals of each other, find k.
  2. 8If the zeroes of x² + (k − 3)x − 3 are equal in magnitude but opposite in sign, find k.
Show answers (try them yourself first)
  1. 13
  2. 26
  3. 38
  4. 4−5
  5. 56
  6. 6−3
  7. 70
  8. 83

Expressions in α and β

From daily life

You often need α² + β² or 1/α + 1/β. You do not need the actual zeroes: only their sum and product.

Simple definition

Write the expression using α + β and αβ. Key identities: α² + β² = (α + β)² − 2αβ, 1/α + 1/β = (α + β)/αβ, (α − β)² = (α + β)² − 4αβ.

Formula: α² + β² = (α + β)² − 2αβ

α² + β² = (α+β)² − 2αβ1/α + 1/β = (α+β)/αβα³+β³ = S³ − 3PS

In the figure: For x² − 5x + 6: S = 5, P = 6, so α² + β² = 25 − 12 = 13.

Rule 1: Squares and reciprocals — Find S = −b/a and P = c/a first.

  • Easy: x² − 5x + 6: α² + β² = 13, 1/α + 1/β = 5/6.
  • Medium: x² + x − 6: S = −1, P = −6: α² + β² = 1 + 12 = 13.
  • Exam-type: Same quadratic: 1/α + 1/β = (−1)/(−6) = 1/6.

Rule 2: Difference and cubes — Use (α − β)² and α³ + β³ = S³ − 3PS.

  • Easy: x² − 5x + 6: (α − β)² = 25 − 24 = 1.
  • Medium: x² − 3x + 1: α² + β² = 9 − 2 = 7.
  • Exam-type: x² − 3x + 1: α³ + β³ = 27 − 9 = 18.
Remember

You can check each answer by finding the zeroes when they are whole numbers.

Common mistake

Writing α² + β² = (α + β)² + 2αβ with the wrong sign.

Memory clue

"Square the sum, subtract two products."

Practice questions

Easy
  1. 1If α, β are the zeroes of x² + 8x + 7, find the value of 2α + 2β − αβ.
  2. 2If α, β are the zeroes of x² − x − 4, find the value of 2α + 2β + 3αβ.
  3. 3If α, β are the zeroes of x² + 8x + 4, find the value of α + β − αβ.
Medium
  1. 4If α, β are the zeroes of x² − 6x − 9, find the value of 2α + 2β − αβ.
  2. 5If α and β are the zeroes of 3x² + 9x + 3, find the value of α²β + αβ².
  3. 6If α and β are the zeroes of x² − 9x − 3, find the value of 1/α + 1/β.
Challenging
  1. 7If α and β are the zeroes of x² − 3x − 9, find the value of α/β + β/α.
  2. 8If α and β are the zeroes of 3x² + 7x − 9, find the value of 1/α + 1/β.
Show answers (try them yourself first)
  1. 1−23
  2. 2−10
  3. 3−12
  4. 421
  5. 5−3
  6. 6−3
  7. 7−3
  8. 87/9