Question

Find a quadratic polynomial whose zeroes are $(5 - 2\sqrt{3})$ and $(5 + 2\sqrt{3})$.

Hint:

Notice that irrational zeroes of polynomials with rational coefficients always occur in conjugate pairs ($a - \sqrt{b}$ and $a + \sqrt{b}$).
This guarantees that their sum and product are always rational numbers.

Answer 1

Step 1: Build the polynomial directly from its factors.
If the zeroes are $5-2\sqrt3$ and $5+2\sqrt3$, the polynomial can be written as $p(x)=\big(x-(5-2\sqrt3)\big)\big(x-(5+2\sqrt3)\big)$.
Step 2: Group terms to use the difference-of-squares pattern.
\[ p(x) = \big[(x-5)+2\sqrt3\big]\big[(x-5)-2\sqrt3\big] = (x-5)^2-(2\sqrt3)^2 \]
Step 3: Expand and simplify.
\[ p(x) = (x^2-10x+25) - 12 = x^2-10x+13 \]
\[ \boxed{x^2-10x+13} \]