IB Maths AA HL Topic 2 — Functions Paper 1 & 2 HL only ~7 min read

Sum & Product of Roots of Polynomials

The sum and product of all the roots of a polynomial sit hidden inside just two of its coefficients. You don’t need to factorise — you can read both off directly from the formula. That makes them powerful when the question gives you partial information about the roots and wants you to find the rest, or to back out an unknown coefficient.

📘 What you need to know

For P(x) = a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀ with roots α₁, …, α_n:
Sum of roots: α₁ + α₂ + … + α_n = −a_n−1/a_n.
Product of roots: α₁ · α₂ · … · α_n = (−1)ⁿ · a₀/a_n.
Both formulas are in the formula booklet — but you must know how to use them under exam pressure.
Roots include complex ones: real and complex roots both count toward the sum and product. For real polynomials, complex roots come in conjugate pairs (which always sum and multiply to real values).
Repeated roots count multiple times: a double root contributes twice to the sum and twice to the product.
Watch for missing terms: if a coefficient is missing from the polynomial, treat it as 0 (don’t skip it).

The two formulas

Sum of roots α₁ + α₂ + … + α_n = −a_n−1a_n ✓ in formula booklet

Product of roots α₁ · α₂ · … · α_n = (−1)ⁿ · a₀a_n ✓ in formula booklet

Sum — what you need

leading coeff a_n
next coeff a_n−1

divide the second by the first, flip sign

Product — what you need

leading coeff a_n
constant a₀
degree n

divide constant by leading; flip sign if n is odd

Quick check on signs: for a monic polynomial (a_n = 1), sum = −a_n−1 and product = (−1)ⁿ·a₀. Cubics flip the constant’s sign for product; quartics keep it. Always count the degree carefully.

Watch out for missing terms

If the polynomial is missing a power of x (no x³ term, no constant, etc.), the corresponding coefficient is zero, not skipped.

Example with placeholders x⁴ + 3x² − 5 = x⁴ + 0x³ + 3x² + 0x − 5

So a₃ = 0 (sum of roots = 0) and a₀ = −5. The same trick that helped in polynomial division — explicitly inserting placeholders — prevents most slips here.

Complex roots in the sum and product

For a real polynomial, complex roots come in conjugate pairs. The pair (a + bi) and (a − bi) contribute clean real values:

Conjugate pair shortcut sum: (a + bi) + (a − bi) = 2a · product: (a + bi)(a − bi) = a² + b²

When working with complex roots, always pair them with their conjugates first. The pair gives you a real sum and a real product — no imaginary algebra to track. The total sum and product over all roots are then simple addition and multiplication.

🧭 Recipe — using sum/product to find unknowns

List all the roots you know. Add the conjugate of any complex root automatically (real polynomials only).
Compute the sum of known roots.
Compute the product of known roots — pair conjugates first to keep things real.
Apply each formula: set sum equal to −a_n−1/a_n; set product equal to (−1)ⁿ·a₀/a_n.
Solve the resulting equation(s) for the unknown(s).

Worked examples

WE 1

Read off sum and product of roots

Find the sum and product of the roots of 4x³ − 7x² + 2x − 9 = 0.

Identify the coefficients (n = 3) a₃ = 4, a₂ = −7, a₀ = −9 Sum = −a₂/a₃ sum = −(−7)/4 = 7/4 Product = (−1)³ · a₀/a₃ product = (−1) · (−9)/4 = 9/4 sum = 7/4; product = 9/4 no factorisation, no roots — just two coefficient picks

WE 2

Polynomial with missing terms

Find the sum and product of the roots of x⁴ + 3x² − 5 = 0.

Insert placeholders for missing terms x⁴ + 0x³ + 3x² + 0x − 5 a₄ = 1, a₃ = 0, a₀ = −5 Sum = −a₃/a₄ sum = −0/1 = 0 Product = (−1)⁴ · a₀/a₄ product = 1 · (−5)/1 = −5 sum = 0; product = −5 missing x³ term gives sum 0 — easy to overlook if you skip the placeholder

WE 3

Find an unknown coefficient given the sum of roots

The roots of 2x³ + ax² − 6x + 12 = 0 sum to 5. Find the value of a.

Apply sum formula sum = −a₂/a₃ = −a/2 = 5 −a/2 = 5 → a = −10 a = −10 one equation, one unknown — sum alone is enough

WE 4

Find unknowns from sum and product

The cubic 2x³ + bx² − 11x + d = 0 has roots whose sum is 3 and whose product is −8. Find b and d.

Apply sum formula: −b/2 = 3 b = −6 Apply product formula: (−1)³ · d/2 = −8 −d/2 = −8 → d = 16 b = −6; d = 16 sum and product give two independent equations — perfect for two unknowns

WE 5

Find an unknown root using the sum formula

The equation 4x⁴ + 8x³ + … = 0 has roots α (real), 1, 2 + i and 2 − i. Find α.

Apply sum formula sum = −a₃/a₄ = −8/4 = −2 Sum the known roots α + 1 + (2 + i) + (2 − i) = α + 5 Set equal and solve α + 5 = −2 → α = −7 α = −7 complex pair (2 + i) + (2 − i) = 4 — always real for conjugate pairs

WE 6

Find a real root and an unknown constant

The equation 2x⁵ − 10x⁴ + ax³ + bx² + cx + k = 0 has real coefficients and roots that include 1 + 2i, i, and a real root α.
(a) Find α. (b) Find k.

Step 1: Use conjugate root theorem real polynomial → 1 − 2i and −i are also roots five roots: 1 + 2i, 1 − 2i, i, −i, α (a) Sum formula: −a₄/a₅ = −(−10)/2 = 5 (1 + 2i) + (1 − 2i) + i + (−i) + α = 2 + 0 + α = 2 + α 2 + α = 5 → α = 3 (a) α = 3 (b) Product formula: (−1)⁵ · k/2 = −k/2 (1 + 2i)(1 − 2i) = 1 + 4 = 5 (i)(−i) = −i² = 1 product of roots = 5 · 1 · 3 = 15 −k/2 = 15 → k = −30 (b) k = −30 pairing conjugates first — (1 + 2i)(1 − 2i) = 5 — keeps everything real, no imaginary algebra

💡 Top tips

Both formulas are in the booklet — but you’ll lose marks if you can’t apply them quickly. Practice until reading off coefficients is automatic.
Always insert placeholders for missing terms before reading off a_n−1.
Pair complex conjugates first: their sum is 2a, their product is a² + b² — both real.
Sum gives one equation, product gives another. Two unknowns? Use both.
Watch the parity of n in the product formula. Odd degree flips the sign of a₀/a_n; even degree doesn’t.
For real polynomials, automatically add complex conjugates to your list of roots. Forgetting them throws off both the sum and the product.
Use sum/product as a sanity check after factorising — the sum of your roots should equal −a_n−1/a_n, the product should equal (−1)ⁿ·a₀/a_n.

⚠ Common mistakes

Skipping placeholder zeros for missing terms — leads to using the wrong coefficient as a_n−1.
Forgetting the (−1)ⁿ factor in the product formula. Cubics need a sign flip; quintics too.
Forgetting the negative in the sum formula. The minus sign is essential — −a_n−1/a_n, not a_n−1/a_n.
Listing one of a complex pair. For real polynomials, both a + bi and a − bi are roots — count both.
Forgetting repeated roots. A double root contributes twice; a triple root contributes three times.
Using the wrong leading coefficient: a_n is the coefficient of the highest power, not the first one written.
Multiplying complex roots without pairing them first — the imaginary algebra wastes time and invites mistakes.

And that closes Section 2.6 — Polynomial Functions. You’ve got the full toolkit now: division, factor and remainder theorems, sketching from factored form, and Vieta’s formulas for sum and product. The next section, Inequalities, applies that polynomial machinery to find the regions where a polynomial is positive or negative — useful for solving things like x³ > 2x or finding the domain of a function involving a square root.

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