Simplify each expression and state any non-permissible values. (5 marks - show your work)
3x+1/2x^2-2 + 2x+2/2x^2-8x+6 is the expression
Can you help please?
2x^2-2 = 2(x-1)(x+1)
2x^2-8x+6 = 2(x-1)(x-3)
Now rewrite your fractions with the factors, put all over a common denominator of 2(x-1)(x+1)(x-3) and simplify the numerator. Probably something will cancel out.
Non-permissible values are those that make any denominator zero.
Of course! To simplify the given expression, we need to combine like terms and perform any necessary operations.
Let's start by grouping the terms with the same denominator. We have:
(3x + 1)/(2x^2 - 2) + (2x + 2)/(2x^2 - 8x + 6)
Now, let's find the least common denominator (LCD) of the two fractions. The LCD is the product of the denominators, which in this case is (2x^2 - 2)(2x^2 - 8x + 6).
Now, we need to rewrite each fraction with the LCD as the denominator. To do this, multiply the numerator and denominator of each fraction by the missing factors. Applying this to the first fraction, we get:
[(3x + 1)(2x^2 - 8x + 6)] / [(2x^2 - 2)(2x^2 - 8x + 6)]
Similarly, for the second fraction:
[(2x + 2)(2x^2 - 2)] / [(2x^2 - 2)(2x^2 - 8x + 6)]
Now, we can combine the fractions by adding their numerators:
[(3x + 1)(2x^2 - 8x + 6) + (2x + 2)(2x^2 - 2)] / [(2x^2 - 2)(2x^2 - 8x + 6)]
Next, let's expand and simplify the numerators:
[6x^3 - 12x^2 + 8x + 2x^2 - 4x + 3 + 4x^3 - 4x^2 + 2x + 2] / [(2x^2 - 2)(2x^2 - 8x + 6)]
Combining like terms, we get:
[10x^3 - 8x^2 + 8x + 5] / [(2x^2 - 2)(2x^2 - 8x + 6)]
Lastly, we can factor the denominator if possible. In this case, we can factor out a common factor of 2:
[10x^3 - 8x^2 + 8x + 5] / [2(x^2 - 1)(2x^2 - 8x + 6)]
Now, we have simplified the expression.
As for the non-permissible values, we need to identify any values of x that would make the denominator equal to zero. In this case, the denominator (2x^2 - 2)(2x^2 - 8x + 6) will be zero if any of the factors are zero.
Setting each factor equal to zero, we have:
x^2 - 1 = 0
x = ±1
2x^2 - 8x + 6 = 0
Unfortunately, this quadratic equation doesn't factor easily. We can solve it using the quadratic formula:
x = (-(-8) ± √((-8)^2 - 4(2)(6))) / (2(2))
x = (8 ± √(64 - 48)) / 4
x = (8 ± √16) / 4
x = (8 ± 4) / 4
x = 3 or x = 1/2
Therefore, the non-permissible values for the expression are x = -1, 1/2, and 1.
I hope this helps! Let me know if you have any further questions.