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Lesson 3 of 4

Polynomials · Lesson 3 of 4

Relationship between Zeroes and Coefficients of a Polynomial

“Where algebra meets matchmaking: discovering the perfect mathematical harmony between the roots of an equation and its coefficients.”

Learning Objectives

• Understand how the zeroes of a polynomial are connected to its coefficients. • Derive and use the relationships for quadratic polynomials. • Derive and use the relationships for cubic polynomials. • Find the sum and product of zeroes without solving the polynomial completely. • Form a quadratic polynomial when its zeroes are known. • Check answers using the relationships between zeroes and coefficients.

Relationship Between Zeroes and Coefficients

The zeroes of a polynomial and its coefficients are closely connected. This connection allows us to find the sum and product of zeroes without calculating each zero separately. It also helps us form a polynomial when its zeroes are given.

Core Idea

For a polynomial written in standard form, the sum and product of its zeroes can be found directly from its coefficients.

Before using any formula, write the polynomial in descending powers of x and identify its coefficients carefully. Include zero as the coefficient of any missing term.

Identifying Missing Coefficients

Problem
Identify a, b and c in 3x² − 12.

  1. 1.Write the polynomial as 3x² + 0x − 12.
  2. 2.Compare it with ax² + bx + c.
  3. 3.Therefore, a = 3, b = 0 and c = −12.

Quadratic Polynomial

Consider a quadratic polynomial p(x) = ax² + bx + c, where a ≠ 0. Suppose its zeroes are α and β.

Factor Form of a Quadratic PolynomialLaTeX
If α and β are the zeroes, then x − α and x − β are factors of the polynomial.
Deriving the RelationshipsLaTeX
Expand the factors and compare the coefficients with ax² + bx + c.

Comparing ax² − a(α + β)x + aαβ with ax² + bx + c gives −a(α + β) = b and aαβ = c.

Zeroes and Coefficients of a Quadratic PolynomialLaTeX
These formulas apply when the polynomial is written as ax² + bx + c with a ≠ 0.
Quick Memory Rule

Sum of zeroes → negative coefficient of x divided by coefficient of x². Product of zeroes → constant term divided by coefficient of x².

Worked Example 1: Finding Sum and Product

Problem
Find the sum and product of the zeroes of 2x² − 7x + 3.

  1. 1.Compare 2x² − 7x + 3 with ax² + bx + c.
  2. 2.Here, a = 2, b = −7 and c = 3.
  3. 3.Sum of zeroes = −b/a = −(−7)/2 = 7/2.
  4. 4.Product of zeroes = c/a = 3/2.
  5. 5.Therefore, α + β = 7/2 and αβ = 3/2.
CalculationLaTeX
Be careful: b is −7, so −b becomes positive 7.
Worked Example 2: Missing Middle Term

Problem
Find the sum and product of the zeroes of 3x² − 12.

  1. 1.Write the polynomial as 3x² + 0x − 12.
  2. 2.Here, a = 3, b = 0 and c = −12.
  3. 3.Sum of zeroes = −0/3 = 0.
  4. 4.Product of zeroes = −12/3 = −4.
  5. 5.Therefore, the sum is 0 and the product is −4.
Worked Example 3: Verifying by Factorisation

Problem
Verify the relationship for p(x) = x² − 5x + 6.

  1. 1.Factorise: x² − 5x + 6 = (x − 2)(x − 3).
  2. 2.Therefore, the zeroes are 2 and 3.
  3. 3.From the zeroes, sum = 2 + 3 = 5 and product = 2 × 3 = 6.
  4. 4.From the coefficients, a = 1, b = −5 and c = 6.
  5. 5.−b/a = −(−5)/1 = 5 and c/a = 6/1 = 6.
  6. 6.Both methods give the same results.
Watch the Sign of b

The formula for the sum contains a minus sign: α + β = −b/a. Always substitute the sign of b correctly. For example, if b = −5, then −b = 5.

Forming a Quadratic Polynomial from Its Zeroes

If the zeroes α and β are known, a quadratic polynomial having these zeroes can be formed using the factors x − α and x − β.

Polynomial with Zeroes α and βLaTeX
Multiplying by any non-zero constant gives another polynomial with the same zeroes.
Simplest Quadratic PolynomialLaTeX
Choosing k = 1 gives the simplest polynomial.
Worked Example 4: Forming a Polynomial from Two Zeroes

Problem
Form a quadratic polynomial whose zeroes are 3 and −2.

  1. 1.The zeroes are α = 3 and β = −2.
  2. 2.Use p(x) = (x − α)(x − β).
  3. 3.p(x) = (x − 3)(x + 2).
  4. 4.Expand: p(x) = x² − x − 6.
  5. 5.Therefore, one required polynomial is x² − x − 6.
Worked Example 5: Using Sum and Product

Problem
Form a quadratic polynomial whose sum of zeroes is 7 and product of zeroes is 10.

  1. 1.Use x² − (sum of zeroes)x + product of zeroes.
  2. 2.Substitute the given sum and product.
  3. 3.p(x) = x² − 7x + 10.
  4. 4.Therefore, one required polynomial is x² − 7x + 10.
Worked Example 6: Fractional Zeroes

Problem
Form a polynomial whose zeroes are 1/2 and 3.

  1. 1.Start with p(x) = (x − 1/2)(x − 3).
  2. 2.Expanding gives x² − (7/2)x + 3/2.
  3. 3.To remove fractions, multiply the entire polynomial by 2.
  4. 4.Therefore, 2x² − 7x + 3 is a polynomial with the required zeroes.
Removing Fractional CoefficientsLaTeX
Multiplying by 2 does not change the zeroes.

Cubic Polynomial

Now consider a cubic polynomial p(x) = ax³ + bx² + cx + d, where a ≠ 0. Suppose its zeroes are α, β and γ.

Factor Form of a Cubic PolynomialLaTeX
Each zero produces one factor.
Expansion of the Cubic Factor FormLaTeX
The middle relationship uses the sum of the products of the zeroes taken two at a time.
Zeroes and Coefficients of a Cubic PolynomialLaTeX
Compare the expanded expression with ax³ + bx² + cx + d.
Cubic Memory Rule

For ax³ + bx² + cx + d: Sum of zeroes = −b/a Sum of products taken two at a time = c/a Product of all three zeroes = −d/a

Worked Example 7: Cubic Relationships

Problem
For p(x) = 2x³ − 3x² − 8x + 12, find the three coefficient relationships.

  1. 1.Compare with ax³ + bx² + cx + d.
  2. 2.Here, a = 2, b = −3, c = −8 and d = 12.
  3. 3.α + β + γ = −b/a = −(−3)/2 = 3/2.
  4. 4.αβ + βγ + γα = c/a = −8/2 = −4.
  5. 5.αβγ = −d/a = −12/2 = −6.
Worked Example 8: Verifying a Cubic Polynomial

Problem
Verify the relationships for p(x) = x³ − 6x² + 11x − 6, whose zeroes are 1, 2 and 3.

  1. 1.Sum of zeroes = 1 + 2 + 3 = 6.
  2. 2.From coefficients, −b/a = −(−6)/1 = 6.
  3. 3.Sum of products taken two at a time = (1×2) + (2×3) + (3×1) = 11.
  4. 4.From coefficients, c/a = 11.
  5. 5.Product of zeroes = 1 × 2 × 3 = 6.
  6. 6.From coefficients, −d/a = −(−6) = 6.
  7. 7.All three relationships are verified.
Meaning of Pairwise Products

The expression αβ + βγ + γα means multiply the zeroes in pairs and then add the three results.

PolynomialSum RelationshipOther Relationships
Quadratic: ax² + bx + cα + β = −b/aαβ = c/a
Cubic: ax³ + bx² + cx + dα + β + γ = −b/aαβ + βγ + γα = c/a; αβγ = −d/a

How to Solve These Questions

  1. Write the polynomial in descending powers of x.
  2. Insert zero coefficients for missing terms.
  3. Identify a, b, c and d with their signs.
  4. Choose the correct quadratic or cubic formula.
  5. Substitute carefully, especially when a coefficient is negative.
  6. Simplify the fractions.
  7. When possible, verify using factorisation.
Common Mistakes

• Forgetting the negative sign in the sum formula. • Using b/a instead of −b/a. • Ignoring the sign of a coefficient while substituting. • Forgetting that a missing term has coefficient zero. • Using c instead of c/a for the product of quadratic zeroes. • For a cubic polynomial, confusing αβ + βγ + γα with α + β + γ. • Assuming that only one polynomial can have a given set of zeroes.

Quiz

Quick check

For ax² + bx + c, what is the sum of its zeroes?

Quick check

For ax² + bx + c, what is the product of its zeroes?

Quick check

What is the sum of the zeroes of x² − 5x + 6?

Quick check

Which polynomial has zeroes 3 and −2?

Quick check

For ax³ + bx² + cx + d, what is the product αβγ?

Practice Problems

Practice Questions
  1. Find the sum and product of the zeroes of x² − 8x + 15.
  2. Find the sum and product of the zeroes of 3x² + 5x − 2.
  3. Find the sum and product of the zeroes of 4x² − 9.
  4. Verify the relationships for x² + x − 6 by first finding its zeroes.
  5. Form a quadratic polynomial whose zeroes are 4 and 5.
  6. Form a quadratic polynomial whose zeroes are −3 and 2.
  7. Form a quadratic polynomial whose sum of zeroes is 6 and product is 8.
  8. Form a polynomial whose zeroes are 2/3 and −1.
  9. For 2x³ + 5x² − 4x − 3, find the three coefficient relationships.
  10. Verify the relationships for x³ − 3x² − 4x + 12 if its zeroes are −2, 2 and 3.

Key Takeaways

Key Takeaways

• For ax² + bx + c, the sum of zeroes is −b/a and their product is c/a. • A quadratic polynomial with zeroes α and β can be written as k(x − α)(x − β), where k ≠ 0. • The simplest polynomial with sum S and product P is x² − Sx + P. • For ax³ + bx² + cx + d, the sum of zeroes is −b/a. • For a cubic polynomial, the sum of pairwise products is c/a and the product of all three zeroes is −d/a. • Always identify coefficients with their signs and include zero coefficients for missing terms.

Previous · Lesson 2

Understand the geometrical meaning of a zero

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Chapter Summary and Practice