Perform the following operations and prove closure. Show your work.
x over x+3 + x+2 over x+5
x+4 over x^2+5x+6 • x+3 over x^2−16
2 over x^2−9 − 3x over x^2−5x+6
x+4 over x^2−5x+6 / x^2−16 over x+3
Compare and contrast division of integers to division of rational expressions.
Next time please type your problems in a more normal way
I will do the 3rd:
2/(x^2−9) − 3x/(x^2−5x+6)
step1: try to factor
= 2/((x+3)(x-3)) - 3x/((x-2)(x-3))
so the LCD is (x+3)(x-3)(x-2)
= (2(x-2) - 3x(x+3) )/( (x+3)(x-3)(x-2))
= (2x - 4 - 3x^2 - 3x) / ( (x+3)(x-3)(x-2))
= -(3x^2 + x + 4) / ( (x+3)(x-3)(x-2))
there should have been a restriction stated throughout, namely
x ≠ ± 3, 2
To perform the given operations and prove closure, we need to simplify each expression and show that the result is still a rational expression with a defined denominator for all valid values of x.
1. (x / (x+3)) + ((x+2) / (x+5))
To combine the fractions, we need a common denominator, which is (x+3)(x+5). After obtaining the common denominator, we can add the fractions:
((x)(x+5) + (x+2)(x+3)) / ((x+3)(x+5))
Expanding and simplifying the numerator:
(x^2 + 5x + x^2 + 5x + 6 + 3x + 9) / ((x+3)(x+5))
(2x^2 + 13x + 15) / ((x+3)(x+5))
We can see that the denominator is defined and not equal to zero for all valid values of x. Therefore, closure is proven for this expression.
2. ((x+4) / (x^2 + 5x + 6)) * ((x+3) / (x^2 - 16))
Multiplying rational expressions is similar to multiplying fractions. We multiply the numerators together and the denominators together:
((x+4)(x+3)) / ((x^2 + 5x + 6)(x^2 - 16))
Expanding and simplifying the numerator and denominator:
(x^2 + 7x + 12) / ((x + 2)(x + 3)(x - 4)(x + 4))
We can see that the denominator is defined and not equal to zero for all valid values of x. Therefore, closure is proven for this expression.
3. (2 / (x^2 - 9)) - ((3x) / (x^2 - 5x + 6))
To subtract these rational expressions, we again need a common denominator, which is (x^2 - 9)(x^2 - 5x + 6). After obtaining the common denominator, we can subtract the fractions:
((2)(x^2 - 5x + 6) - (3x)(x^2 - 9)) / ((x^2 - 9)(x^2 - 5x + 6))
Expanding and simplifying the numerator:
(2x^2 - 10x + 12 - 3x^3 + 27x) / ((x^2 - 9)(x^2 - 5x + 6))
(-3x^3 + 2x^2 + 17x + 12) / ((x^2 - 9)(x^2 - 5x + 6))
We can see that the denominator is defined and not equal to zero for all valid values of x. Therefore, closure is proven for this expression.
4. ((x+4) / (x^2 - 5x + 6)) / ((x^2 - 16) / (x+3))
To divide rational expressions, we multiply the first fraction by the reciprocal of the second. In this case, we multiply by ((x+3) / (x^2 - 16)):
((x+4) / (x^2 - 5x + 6)) * ((x+3) / (x^2 - 16))
Expanding and simplifying the numerator and denominator:
(x^2 + 7x + 12) / ((x - 2)(x - 3)(x + 4)(x + 3))
We can see that the denominator is defined and not equal to zero for all valid values of x. Therefore, closure is proven for this expression.
Now, let's compare and contrast the division of integers to the division of rational expressions:
Division of Integers:
- In division of integers, we perform division of whole numbers or integers without fractional parts.
- The quotient is either an integer or a fraction.
- Division by zero is undefined.
- Closure is guaranteed, meaning the division of two integers will always result in another integer or a fraction.
Division of Rational Expressions:
- In division of rational expressions, we perform division of algebraic expressions (polynomials) that have fractional coefficients.
- The quotient is always a rational expression, which is a fraction of polynomials.
- Division by zero in the denominator is undefined, but division by zero in the numerator is allowed unless it leads to division by zero in the denominator.
- Closure is not always guaranteed since the resulting expression may have restrictions on the denominator, where certain values of the variable make the expression undefined.
In summary, division of integers and division of rational expressions have some similarities, such as the concept of division and the potential presence of fractions. However, they also differ in terms of closure, treatment of division by zero, and the resulting quotient.