Write a polynomial function of least degree with integral coefficents that has given zeros ¾, -3, -2/5?

Answer 1

Write a polynomial function of least degree with integral coefficents that has given zeros ¾, -3, -2/5?

Since the polynomial function will have three roots, we can expect the answer to be a cubic equation.

Here's how to find the cubic polynomial..

If (4x - 3) is a factor of the cubic equation, then x = 3/4 will be a zero (root).

If (x + 3) is a factor of the cubic equation, then x = -3 will be a zero (root).

If (5x + 2) is a factor of the cubic equation, then x = -2/5 will be a zero (root).

All that is left is to multiply the three factors to obtain a cubic polynomial with the desired roots.

Tackling the more difficult multiplication first, (4x - 3)(5x + 2) = 20x² + 8x - 15x - 6 = 20x² - 7x - 6.

Now multiplying that quadratic by (X + 3),

(X + 3)(20x² - 7x - 6) = 20x³ -7x² - 6x + 60x² - 21x - 18.

Combining like terms, it simplifies to 20x³ + 53x² - 27x - 19. Since the coefficients have not factors in common, it is lowest terms.

So the final solution is 20x³ + 53x² - 27x - 19 = 0.