Question

Which of the following is a polynomial with roots: − square root of 3 , square root of 3, and −2? (6 points) Question 7 options: 1) x3 − 2x2 − 3x + 6 2) x3 − 3x2 − 5x + 15 3) x3 + 2x2 − 3x − 6 4) x3 + 3x2 − 5x − 15

Answer 1

Answer:

The polynomial is $x^{3} - 1.46x^{2} - 3.93x + 6$

Step-by-step explanation:

A nth order polynomial f(x) has roots $x_{1}, x_{2}, ..., x_{n}$ such that $f(x) = (x - x_{1})*(x - x_{2})*...*(x - x_{n}}$,

Which of the following is a polynomial with roots: − square root of 3 , square root of 3, and −2?

So

$x_{1} = x_{2} = \sqrt{3}$

$x_{3} = -2$

Then

$(x - \sqrt{3})^{2}*(x - (-2)) = (x - \sqrt{3})^{2}*(x + 2) = (x^{2} -2x\sqrt{3} + 3)*(x + 2) = x^{3} + 2x^{2} - 2x^{2}\sqrt{3} - 4x\sqrt{3} + 3x + 6$

Since $\sqrt{3} = 1.73$

$x^{3} + 2x^{2} - 3.46x^{2} - 6.93x + 3x + 6 = x^{3} - 1.46x^{2} - 3.93x + 6$

The polynomial is $x^{3} - 1.46x^{2} - 3.93x + 6$