4 This is hard for me to understand how to work them...
What is the simplified form of the product
[(x^2 + 8x + 15)/(x-4)] * [(x^2 – 16) / 2x + 6)]
x^2+8x+15 = (x+3)(x+5)
x^2-16 = (x+4)(x-4)
so, what you have is
(x+3)(x+5) / (x-4) * (x-4)(x+4) / 2(x+3)
Now you see that x+3 and x-4 cancel out, leaving
(x+5)(x-4)/2
so ..
[(x^2 + 8x + 15)/(x-4)] * [(x^2 – 16) / (2x + 6)]
= (x+3)(x+5)/(x-4) * (x-4)(x+4)/(2(x+3) )
= (x+5)(x+4)/2 , x ≠ -3, 4
the temptation is to anticipate asymptotes at x = -3 and x = 4
but since we obtain 0/0 for these two values in the original, but an actual value for the final simplified form, we get a hole for these two values of x
Thus the restriction must be part of the answer.
To find the simplified form of the product, we need to simplify each fraction separately and then multiply them together.
Let's simplify the first fraction: (x^2 + 8x + 15)/(x - 4)
To simplify this fraction, we need to factor the numerator and denominator if possible. Factoring the numerator, we have:
x^2 + 8x + 15 = (x + 3)(x + 5)
Now let's factor the denominator: x - 4
Since both the numerator and denominator are in factored form, we can cancel out any common factors. In this case, we have (x + 5) in the numerator and the denominator, so we can cancel them out:
(x^2 + 8x + 15)/(x - 4) = (x + 3)
Now let's simplify the second fraction: (x^2 - 16)/(2x + 6)
Again, we need to factor the numerator and denominator. The numerator is a difference of squares:
x^2 - 16 = (x - 4)(x + 4)
And the denominator can also be factored by taking out the common factor of 2:
2x + 6 = 2(x + 3)
Now, let's cancel out any common factors:
(x^2 - 16)/(2x + 6) = (x - 4)/(x + 3)
Now, multiply the two simplified fractions together:
[(x + 3) * (x - 4)] / [(x + 3)] = (x - 4)
Therefore, the simplified form of the product is (x - 4).