Build it from sum and product
Sum and product fix the quadratic, up to k.
Rebuild it
Someone says only this: the zeroes of a quadratic add up to −3 and multiply to 2. Can you write the polynomial?
Which polynomial is it?
What this lesson covers
The idea
A quadratic polynomial with zeroes α and β is k(x – α)(x – β) = k[x² – (α + β)x + αβ] for a constant k, so the sum and product of its zeroes fix it up to k.
Rebuild it
Someone says only this: the zeroes of a quadratic add up to −3 and multiply to 2. Can you write the polynomial?
Which polynomial is it?
- x² + 3x + 2
- x² − 3x + 2
- x² + 3x − 2
Build it
Set the sum and the product. The machine writes the polynomial and checks its zeroes.
k(x − α)(x − β)
A quadratic polynomial with zeroes α and β is k(x − α)(x − β) = k[x² − (α + β)x + αβ], where k is a constant.
So the sum and the product of the zeroes fix the polynomial, up to k. Example: sum −3 and product 2 give x² + 3x + 2. Any other such polynomial is k(x² + 3x + 2).
Notes
A quadratic with zeroes α and β is k[x² − (α + β)x + αβ]. Sum and product fix it up to the constant k.
Check yourself
A quadratic x² + bx + c has zeroes that add to 5 and multiply to 6. Find c.
Answer: 6
x² − (sum)x + (product) = x² − 5x + 6. So c = 6.
Same polynomial: sum 5, product 6, first coefficient 1. Find b.
Answer: -5
b = −(sum) = −5. The polynomial is x² − 5x + 6, with zeroes 2 and 3.
Which quadratic polynomial has the zeroes 2 and 3?
Do x² + 3x + 2 and 5x² + 15x + 10 have the same zeroes?
- x² − 5x + 6 — correct. Yes! Sum 5, product 6: x² − 5x + 6 = (x − 2)(x − 3).
- x² + 5x + 6. This has the zeroes −2 and −3.
- x² − 6x + 5. This has the zeroes 1 and 5.
- Yes, the second is 5 times the first — correct. Yes! 5(x² + 3x + 2) = 5x² + 15x + 10. Multiplying by 5 does not change when it is 0.
- No, the coefficients are different. The coefficients differ, but 5x² + 15x + 10 = 5(x² + 3x + 2), so the zeroes are the same.
- Only one zero is the same. Both zeroes are the same: −1 and −2.