If point  P(x, y) is such that  y  =  x3 − 2x2 − 7x + 5&x  =  2 + √3 then point  P(x, y) is?

To find the coordinates of point $P(x, y)$ given the equation $y = x^3 - 2x^2 - 7x + 5$ and $x = 2 + \sqrt{3}$, we will follow these steps:

1. Break Down the Problem

We need to substitute $x = 2 + \sqrt{3}$ into the equation for $y$.

2. Relevant Concepts

Recall that to find $y$, we need to evaluate the polynomial at the given value for $x$: $y = x^3 - 2x^2 - 7x + 5$

3. Analysis and Detail

First, calculate $x^2$ and $x^3$:

• Calculate $x^2$: $x^2 = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}$

• Calculate $x^3$: $x^3 = (2 + \sqrt{3})(7 + 4\sqrt{3}) = 14 + 8\sqrt{3} + 7\sqrt{3} + 12 = 26 + 15\sqrt{3}$

Now substitute $x = 2 + \sqrt{3}$, $x^2 = 7 + 4\sqrt{3}$, and $x^3 = 26 + 15\sqrt{3}$ into the equation for $y$: $y = (26 + 15\sqrt{3}) - 2(7 + 4\sqrt{3}) - 7(2 + \sqrt{3}) + 5$

Calculating each term:

• $-2(7 + 4\sqrt{3}) = -14 - 8\sqrt{3}$
• $-7(2 + \sqrt{3}) = -14 - 7\sqrt{3}$

Putting it all together: $y = (26 + 15\sqrt{3}) - (14 + 8\sqrt{3}) - (14 + 7\sqrt{3}) + 5$ Combine like terms: $y = 26 - 14 - 14 + 5 + (15\sqrt{3} - 8\sqrt{3} - 7\sqrt{3})$ $y = 3 + 0\sqrt{3} = 3$

4. Verify and Summarize

After calculating:

• The value of $y$ is $3$ when $x = 2 + \sqrt{3}$.

Final Answer

The point $P(x, y)$ is $\left(2 + \sqrt{3}, 3\right)$.