If 3−√7 is a polynomial root, name another root of the polynomial.Explain how you know it must also be a root.

Answer 1

To find another root of the polynomial with 3−√7 as a root, we can start by assuming that 3−√7 is indeed a root of the polynomial.

We can set up an equation where we substitute x = 3−√7:

P(x) = 0

P(3−√7) = 0

Now, let's manipulate this equation:

P(3−√7) = 0
P(3−√7) = P(x)
0 = P(x)
0 = P(x) - P(x)
0 = P(x) + (-P(x))

By subtracting P(x) from both sides, we obtain:

0 = P(x) + (-P(x))

Since any number added to its negative equals zero, we can rewrite the equation as:

0 = P(x) + (-P(x))
0 = P(x) - P(x)

Now, we can find the roots of this equation:

P(x) - P(x) = 0

Since P(x) - P(x) equals zero, this implies that any value of x is a root of the polynomial. Therefore, any number can be considered as another root of the polynomial when 3−√7 is a root.