Write the product as linear factors: (x^2−4)(x^2−9).
The product (x^2−4)(x^2−9) can be written as linear factors as follows:
(x+2)(x-2)(x+3)(x-3)
Decompose the polynomial x3−2x^2−5x+6 into its linear factors, given that x−1 is a factor.
If x - 1 is a factor of the polynomial x^3 - 2x^2 - 5x + 6, it means that by dividing x^3 - 2x^2 - 5x + 6 by x - 1, we will get a quotient with no remainder.
Performing the long division:
x^2 -x - 6
-----------------
x - 1 | x^3 - 2x^2 - 5x + 6
- (x^3 - x^2)
--------------
- x^2 - 5x
+ (x^2 - x)
-------------
-6x + 6
+(6x - 6)
--------------
0
Therefore, the quotient is x^2 - x - 6.
So, the polynomial x^3 - 2x^2 - 5x + 6 can be factored as (x - 1)(x^2 - x - 6).
To write the product (x^2−4)(x^2−9) as linear factors, we can first factor each quadratic expression using the difference of squares and the difference of cubes formulas.
For the first factor, x^2−4, we can rewrite it as (x+2)(x-2) using the difference of squares formula, where x+2 is one factor and x-2 is the other.
For the second factor, x^2−9, we can rewrite it as (x+3)(x-3) using the difference of squares formula, where x+3 is one factor and x-3 is the other.
Now, we can write the original product as the product of these linear factors:
(x^2−4)(x^2−9) = (x+2)(x-2)(x+3)(x-3).
So, the product (x^2−4)(x^2−9) can be written as the linear factors (x+2)(x-2)(x+3)(x-3).