(cos^2+6cos x + 9) / (cos x + 3)
factoring ... [(cos x + 3)(cos x + 3)] / (cos x + 3)
canceling ... cos x + 3
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To simplify the expression (cos^2 + 6cos x + 9) / (cos x + 3), we can factor the numerator and the denominator separately.
Let's start with the numerator. Notice that it is a quadratic expression in the form of a^2 + 2ab + b^2, where a = cos x and b = 3. Using the formula for factoring a quadratic, we can rewrite the numerator as (a + b)^2:
(cos^2 + 6cos x + 9) = (cos x + 3)^2
Now let's move on to the denominator. It is a simple linear expression, cos x + 3.
The simplified expression becomes:
[(cos x + 3)^2] / (cos x + 3)
Next, we can notice that (cos x + 3) appears in both the numerator and the denominator. To simplify further, we can cancel out this common factor, resulting in:
(cos x + 3)
Therefore, the simplified expression is just (cos x + 3).