Write the product as linear factors: (x^2−4)(x^2−9)
.
To factor the product (x^2 - 4)(x^2 - 9) as linear factors, we first look for common factors and then use difference of squares.
(x^2 - 4) can be factored as (x + 2)(x - 2) since it is a difference of squares:
(x^2 - 4) = (x^2 - 2^2) = (x - 2)(x + 2)
Similarly, (x^2 - 9) can be factored as (x + 3)(x - 3) since it is also a difference of squares:
(x^2 - 9) = (x^2 - 3^2) = (x - 3)(x + 3)
Therefore, the product (x^2 - 4)(x^2 - 9) can be written as:
(x + 2)(x - 2)(x + 3)(x - 3)
To write the product (x^2−4)(x^2−9) as linear factors, we need to factor each quadratic expression separately and then combine the factors. Starting with the first expression x^2−4, we can recognize it as a difference of squares.
(x^2−4) can be factored as (x−2)(x+2).
Moving to the second expression x^2−9, it is a difference of squares as well.
(x^2−9) can be factored as (x−3)(x+3).
Now we can combine the factors from both expressions:
(x−2)(x+2) * (x−3)(x+3)
Therefore, the product (x^2−4)(x^2−9) can be written as linear factors as (x−2)(x+2) * (x−3)(x+3).
To write the product (x^2 − 4)(x^2 − 9) as linear factors, we need to factorize each quadratic expression separately.
We start with the expression x^2 − 4. This expression is the difference of two perfect squares, so we can use the formula a^2 − b^2 = (a + b)(a − b). In this case, a = x and b = 2. Applying the formula, we get:
x^2 − 4 = (x + 2)(x − 2).
Next, we factorize the expression x^2 − 9. Again, this is the difference of two perfect squares. a = x and b = 3 in this case. Applying the formula, we get:
x^2 − 9 = (x + 3)(x − 3).
Now, we can write the product as the multiplication of these linear factors:
(x^2 − 4)(x^2 − 9) = (x + 2)(x − 2)(x + 3)(x − 3).