Polynomials and Equations

Arjun tells Diya: "I'm thinking of two numbers. Their product is 12. Their sum is 7. What are they?"

Diya sets up a single equation. If x is one number, then the other is (7 − x). Their product is x(7 − x) = 12, which gives:

The two secret numbers are 3 and 4. In polynomial language, 3 and 4 are the zeros of p(x) = x² − 7x + 12 — the values that make the polynomial equal zero.

Rahul gets a homework problem: "For p(x) = 3x² − 5x + 2, find the sum and product of the zeros."

Diya shakes her head. There's a shortcut — named after the mathematician Vieta. Look at the coefficients. The sum of zeros is always −b/a and the product is always c/a. No solving required.

That's it. Vieta's formulas read the sum and product directly from the coefficients. This saves enormous time in exams — especially when the question only asks for sum or product, not the individual zeros.

The class is splitting 47 chocolates equally among 5 students. Each student gets 9. But 47 = 5 × 9 + 2, so 2 chocolates are left over — the remainder.

Polynomial division works the same way. When you divide polynomial p(x) by g(x), you get a quotient q(x) and a remainder r(x). The remainder must be "smaller" than the divisor — meaning its degree is less than the degree of g(x).

The remainder is 0, which means (x + 1) divides p(x) exactly — so (x + 1) is a factor. And x = −1 is a zero. This link between remainder, factors, and zeros is the Factor Theorem.