Find the polynomial having roots at $\frac{1}{2}$ , $\frac{1}{3}$ , $- 1$ and $- 2$ .

Answer

The polynomial generated from the given zeros is: $P (x) = 6 x^{4} + 13 x^{3} - 2 x^{2} - 7 x + 2$

Explanation

Step 1: Turn the zeros into factors:

Zero	Factor
$\frac{1}{2}$	$2 x - 1$
$\frac{1}{3}$	$3 x - 1$
$- 1$	$x + 1$
$- 2$	$x + 2$

Step 2: Multiply the factors together:

P (x) = (2 x - 1) \cdot (3 x - 1) \cdot (x + 1) \cdot (x + 2) = 6 x^{4} + 13 x^{3} - 2 x^{2} - 7 x + 2

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Find the polynomial having roots at 21​ , 31​ , −1 and −2 .

The polynomial generated from the given zeros is: P(x)=6x4+13x3−2x2−7x+2

Find the polynomial having roots at $\frac{1}{2}$ , $\frac{1}{3}$ , $- 1$ and $- 2$ .

The polynomial generated from the given zeros is: $P (x) = 6 x^{4} + 13 x^{3} - 2 x^{2} - 7 x + 2$