You have a difference of squares
(x^2-3)(x^2+3)
One of these factors to two real roots, the other to two imaginary roots.
You have a difference of squares
(x^2-3)(x^2+3)
One of these factors to two real roots, the other to two imaginary roots.
Let g(x)= x^4 - 9
wat are the real factors of g(x) and all the real roots of g(x) = 0
To factorize the polynomial g(x) = x^4 - 9, we can use the difference of squares again. Notice that x^4 can be written as (x^2)^2, and 9 can be written as (3)^2.
So, we can rewrite g(x) as (x^2)^2 - (3)^2, which is a difference of squares.
Using the difference of squares formula, we get:
g(x) = (x^2 - 3)(x^2 + 3)
Now, we can identify the factors of g(x) as (x^2 - 3) and (x^2 + 3).
To find the real roots of g(x) = 0, we need to set each factor equal to zero and solve for x:
For the factor x^2 - 3 = 0, adding 3 to both sides:
x^2 = 3
Taking the square root of both sides:
x = ± √3
So, the real roots of the factor x^2 - 3 = 0 are x = √3 and x = -√3.
For the factor x^2 + 3 = 0, subtracting 3 from both sides:
x^2 = -3
Taking the square root of both sides, we encounter an issue because the square root of a negative number is imaginary. Therefore, x^2 + 3 = 0 has no real roots.
In summary, the real factors of g(x) = x^4 - 9 are (x^2 - 3) and (x^2 + 3), and the real roots of g(x) = 0 are x = √3 and x = -√3.
To find the real factors of g(x) = x^4 - 9 and the real roots of g(x) = 0, we can use the difference of squares property.
First, let's factorize g(x) using the difference of squares formula. We have:
g(x) = (x^2)^2 - (3)^2
= (x^2 - 3)(x^2 + 3)
Now, let's find the real factors and roots by setting each factor equal to zero:
Setting x^2 - 3 = 0:
x^2 = 3
x = ±√3 (These are the real roots and factors of this factor)
Setting x^2 + 3 = 0:
x^2 = -3
This equation has no real solutions since a square of any real number cannot be negative. Therefore, this factor has no real roots.
In summary:
The real factor of g(x) = x^4 - 9 is (x^2 - 3). The real roots of g(x) = 0 are x = ±√3. The other factor, (x^2 + 3), has no real roots.