Zeroes talk to coefficients

The idea

If α and β are the zeroes of ax² + bx + c, a ≠ 0, then α + β = –b/a = –(Coefficient of x)/(Coefficient of x²) and αβ = c/a = (Constant term)/(Coefficient of x²).

Add and multiply the zeroes

The zeroes of 2x² − 8x + 6 are 1 and 3. Their sum is 4 and their product is 3. The coefficients are 2, −8 and 6.

Do the sum and product of the zeroes have anything to do with the coefficients?

• No link at all
• Yes, there is a link
• Not sure

Open the table

Tap each polynomial to find its zeroes. Compare the matching colours.

Sum and product

If α and β are the zeroes of ax² + bx + c (a ≠ 0), then α + β = −b/a αβ = c/a

In words: the sum of the zeroes is −(coefficient of x)/(coefficient of x²), and the product is (constant term)/(coefficient of x²).

Why? The factors are x − α and x − β, so ax² + bx + c = k(x − α)(x − β) = kx² − k(α + β)x + kαβ. Compare the coefficients: a = k, b = −k(α + β), c = kαβ. So α + β = −b/a and αβ = c/a.

Check with x² − 3. Its zeroes are √3 and −√3. Their sum is 0, and b = 0, so −b/a = 0. Their product is −3, and c/a = −3/1 = −3.

Notes

If α and β are the zeroes of ax² + bx + c, then α + β = −b/a αβ = c/a

Check yourself

Find the sum of the zeroes of x² − 9x + 20.

Answer: 9

Sum = −b/a = −(−9)/1 = 9. (The zeroes are 4 and 5, and 4 + 5 = 9.)

Find the product of the zeroes of 2x² − 6x + 4.

Answer: 2

Product = c/a = 4/2 = 2. (The zeroes are 1 and 2, and 1 × 2 = 2.)

3x² + 5x − 2 has zeroes 1/3 and −2. What is the product of its zeroes?

The zeroes of x² + bx + 6 add up to 5. Find b.

Answer: -5

Sum = −b/a = −b = 5, so b = −5. The polynomial is x² − 5x + 6, with zeroes 2 and 3.

• −2/3 — correct. Yes! c/a = −2/3, and (1/3) × (−2) = −2/3.
• 2/3. The product is c/a = −2/3. Keep the minus sign.
• −5/3. That is the sum, −b/a = −5/3.
• 5/3. Neither sum nor product: the sum is −5/3 and the product is −2/3.