Concept 2.4
Building a Quadratic from Its Zeroes
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
A quadratic polynomial whose zeroes are α and β is k[x² − (α + β)x + αβ], where k is any non-zero real number.
The intuition
Run the last concept backwards. If you know the sum and the product of the zeroes, you know the polynomial up to a stretch factor k.
Zeroes −1 and 3 have sum 2 and product −3, so x² − 2x − 3 works. Check: (x + 1)(x − 3) = x² − 2x − 3.
The attention
Look at the x-term: it carries the sum with a minus.
k[x² − (α + β)x + αβ]
Look hereNot here
- Look here
- The x-term: minus the sum of zeroes. The x coefficient is minus the sum. Get its sign right and the constant, the product, takes care of itself.
- Not here
- The constant: the product of zeroes. The constant is just αβ, sign and all. Putting a minus there out of habit gives the wrong zeroes.
- What it teaches
- Building backwards is a good check: expand your answer and see if the zeroes come back.