Question

Construct a polynomial of least degree possible with real roots: −4,−1, 1 ,4 and f(−2)=10.

Answer 1

To construct a polynomial with the given real roots, multiply the factors (x + 4), (x + 1), (x - 1), and (x - 4) together. The resulting polynomial is x^4 - 10x^2 + 16.

Step-by-step explanation:

To construct a polynomial with the given real roots, we can use the factor theorem. The polynomial will have factors (x + 4), (x + 1), (x - 1), and (x - 4) since those are the given roots. Multiplying these factors together will give us the polynomial.

(x + 4)(x + 1)(x - 1)(x - 4) = (x^2 + 5x + 4)(x^2 - 5x + 4) = x^4 - 10x^2 + 16

Therefore, the polynomial of least degree with the given real roots is x^4 - 10x^2 + 16.