Write the polynomial expression in the simplest form, x^2 - 9/x^2 - 4 * 4x - 8/12x + 26
The polynomial expression is:
(x^2 - 9) / (x^2 - 4) * (4x - 8) / (12x + 26)
First, let's simplify the expression inside the first parentheses:
(x^2 - 3^2)
This is a difference of squares, which can be factored as:
(x - 3)(x + 3)
Similarly, let's simplify the expression inside the second parentheses:
(4x - 8)
This expression can be factored out a common factor of 4:
4(x - 2)
Lastly, let's simplify the expression inside the second fraction:
(12x + 26)
This expression can be factored out a common factor of 2:
2(6x + 13)
Now, we can substitute these simplified expressions back into the original polynomial expression:
[(x - 3)(x + 3)] / [(x - 2)(x + 2)] * (4(x - 2)) / (2(6x + 13))
Now, let's cancel out any common factors:
(x - 3)(x + 3) / (x - 2)(x + 2) * 2(x - 2) / (6x + 13)
Now, rearrange the expression:
[(x - 3)(x + 3) * 2(x - 2)] / [(x - 2)(x + 2)(6x + 13)]
Multiply the numerators and the denominators:
[2(x - 3)(x + 3)(x - 2)] / [(x - 2)(x + 2)(6x + 13)]
Finally, simplify if possible:
2(x - 3)(x + 3)(x - 2) / (x - 2)(x + 2)(6x + 13)
This is the polynomial expression in its simplest form.
To simplify the given polynomial expression:
Step 1: Factor the denominators:
The numerator (x^2 - 9) can be factored as a difference of squares: (x - 3)(x + 3).
The denominator (x^2 - 4) can be factored as a difference of squares: (x - 2)(x + 2).
The second denominator (12x + 26) can be simplified by factoring out a 2: 2(6x + 13).
Step 2: Simplify the expression:
(x^2 - 9)/(x^2 - 4) * (4x - 8)/(12x + 26)
We can cancel out common factors in the numerator and denominator:
((x - 3)(x + 3))/((x - 2)(x + 2)) * 4(x - 2)/2(6x + 13)
Simplifying further:
(2(x - 3)(x + 3))/(2(x - 2)(x + 2)) * 4(x - 2)/(2(6x + 13))
Now, we can cancel out the common factors of 2 in the numerator and denominator:
((x - 3)(x + 3))/((x - 2)(x + 2)) * 2(x - 2)/(6x + 13)
Simplifying the expression:
2(x - 3)(x + 3)(x - 2)/(6x + 13)
Therefore, the simplified form of the given polynomial expression is:
2(x - 3)(x + 3)(x - 2)/(6x + 13)
To simplify the given expression:
x^2 - 9 / x^2 - 4 * 4x - 8 / 12x + 26
We can start by simplifying each individual fraction separately, and then combine them into a single expression.
Step 1: Simplify the first fraction (x^2 - 9) / (x^2 - 4).
The numerator (x^2 - 9) can be factored as the difference of squares: (x - 3)(x + 3).
The denominator (x^2 - 4) can be factored as the difference of squares: (x - 2)(x + 2).
Therefore, the first fraction can be simplified as: (x - 3)(x + 3) / (x - 2)(x + 2).
Step 2: Simplify the second fraction (4x - 8) / (12x + 26).
Both the numerator and denominator have a common factor of 4, so we can simplify this fraction further.
Dividing both the numerator and denominator by 4, we get: (4(x - 2)) / (4(3x + 7)).
Canceling out the common factor of 4, we have: (x - 2) / (3x + 7).
Now, we have simplified both fractions:
(x - 3)(x + 3) / (x - 2)(x + 2) * (x - 2) / (3x + 7)
The next step is to multiply the numerators and denominators together:
[(x - 3)(x + 3)(x - 2)] / [(x - 2)(x + 2)(3x + 7)]
Next, we can cancel out the common factors of (x - 2) from the numerator and denominator:
(x - 3)(x + 3) / (x + 2)(3x + 7)
Finally, the simplified expression is:
(x^2 - 9) / (3x^2 + 13x + 14)