Cubic: three links
α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.
Zeroes of a cubic
The cubic 2x³ − 5x² − 14x + 8 has zeroes 4, −2 and 12. Their sum is 52. Its coefficients are a = 2, b = −5, c = −14, d = 8.
Which expression gives 52 from the coefficients?
What this lesson covers
The idea
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d, then α + β + γ = –b/a, αβ + βγ + γα = c/a and αβγ = –d/a.
Zeroes of a cubic
The cubic 2x³ − 5x² − 14x + 8 has zeroes 4, −2 and 1/2. Their sum is 5/2. Its coefficients are a = 2, b = −5, c = −14, d = 8.
Which expression gives 5/2 from the coefficients?
- −b/a
- c/a
- −d/a
Open the cubics
Tap each cubic. There are three links to find.
Three links
If α, β, γ are the zeroes of the cubic polynomial ax³ + bx² + cx + d, then α + β + γ = −b/a αβ + βγ + γα = c/a αβγ = −d/a
The sum is −b/a, as for a quadratic. The new link is the sum of the products of the zeroes taken two at a time.
Notes
For the cubic ax³ + bx² + cx + d with zeroes α, β, γ: α + β + γ = −b/a αβ + βγ + γα = c/a αβγ = −d/a
Check yourself
The cubic x³ − 2x² − 5x + 6 has zeroes 1, −2 and 3. Use the coefficients to find α + β + γ.
Answer: 2
−b/a = −(−2)/1 = 2. Check: 1 + (−2) + 3 = 2.
Same cubic: use the coefficients to find αβγ.
Answer: -6
−d/a = −6/1 = −6. Check: 1 × (−2) × 3 = −6.
Which expression equals αβ + βγ + γα?
x³ − 4x has zeroes −2, 0 and 2. Find αβ + βγ + γα using c/a.
Answer: -4
c/a = −4/1 = −4. Check: (−2)(0) + (0)(2) + (2)(−2) = 0 + 0 − 4 = −4.
- c/a — correct. Yes! The sum of the products taken two at a time is c/a.
- −b/a. That is α + β + γ, the sum.
- −d/a. That is αβγ, the product of all three.
- d/a. The product of all three is −d/a, and this is not it.