Writing Polynomial Functions from Real Roots

Question

Which polynomial function has a leading coefficient of –3 and roots –2, √3, and 4, all with multiplicity 1?

Answer 1

With real coefficients, the Irrational Conjugate Root Theorem tells us that if √3 is a root, so is -√3. So the polynomial has 4 roots, -2, √3, -√3, and 4. Use the factored form:

p(x) = a(x-(-2))(x-√3)(x-(-√3))(x-4)

p(x) = -3(x-+2)(x-√3)(x+√3)(x-4)

where a, the leading coefficient, is equal to -3. Since the multiplicity of all roots is 1, then all factors are to the first power. Multiply it out if you want it in standard form.