1. Simplify the expression
4 3
— - —
x x
2. Simplify the expression
6 4
— + —
c c²
3. Simplify the expression
3y 9y
— ÷ —
4y - 8 2y² - 4y
4. Simplify the expression:
( 2x³ - x² - 13x - 6 ) ÷ ( x -3 )
The first one has the common denominator of x, so just subtract the numerators : )
2. needs a common denominator. To make the common denominator of c^2 you must multiply the numerator and denominator of the first fraction by c thus
= 6c/c^2 + 4/c^2
= (6c + 4)/c^2
you could factor out of the top, but it wouldn't change the rational expression : )
3. Factor both denominators and find your common denominator, then continue : )
4. YIPPEEEE! Long division (or synthetic division) your choice : )
Wow you make things really fun Ms Pi!
1. To simplify the expression (4/x) - (3/x), we can combine the terms since they have the same denominator:
(4 - 3) / x = 1/x
So, the simplified expression is 1/x.
2. To simplify the expression (6/c) + (4/c²), we can also combine the terms since they have the same denominator:
(6c + 4) / c²
So, the simplified expression is (6c + 4) / c².
3. To simplify the expression (3y / (4y - 8)) ÷ (9y / (2y² - 4y)), we can simplify each fraction separately and then divide:
For the first fraction, we can factor out 4 from the denominator: 4y - 8 = 4(y - 2)
So, (3y / (4y - 8)) simplifies to 3y / 4(y - 2).
For the second fraction, we can factor out 2y from the denominator: 2y² - 4y = 2y(y - 2)
So, (9y / (2y² - 4y)) simplifies to 9y / 2y(y - 2).
Now, we can divide the two fractions:
(3y / 4(y - 2)) ÷ (9y / 2y(y - 2))
Dividing by a fraction is the same as multiplying by its reciprocal, so we can rewrite the expression as:
(3y / 4(y - 2)) * (2y(y - 2) / 9y)
Simplifying further, we cancel out common factors:
(3y * 2y(y - 2)) / (4(y - 2) * 9y)
= (6y²(y - 2)) / (36y(y - 2))
= (y²(y - 2)) / (6y)
So, the simplified expression is (y²(y - 2)) / (6y).
4. To simplify the expression (2x³ - x² - 13x - 6) ÷ (x - 3), we can use polynomial long division:
2x² + 5x + 4
_____________________
x - 3 | 2x³ - x² - 13x - 6
(2x³ - 6x²)
___________
5x² - 13x
(5x² - 15x)
___________
2x - 6
(2x - 6)
________
0
So, the simplified expression is 2x² + 5x + 4.
To simplify the expressions, we can follow the steps outlined below for each expression:
1. Simplify the expression (4/x) - (3/x):
Since the denominators are the same, we can combine the numerators and keep the common denominator:
(4 - 3)/x = 1/x
Therefore, the simplified expression is 1/x.
2. Simplify the expression (6/c) + (4/c²):
Note that both terms have the common denominator "c" and "c²". We can combine the numerators and keep the common denominator:
(6c + 4)/c²
Therefore, the simplified expression is (6c + 4)/c².
3. Simplify the expression (3y)/(4y - 8) ÷ (9y)/(2y² - 4y):
To divide fractions, we keep the first fraction as it is and multiply by the reciprocal of the second fraction:
(3y)/(4y - 8) * (2y² - 4y)/(9y)
Next, we can simplify the expressions in the numerators and denominators:
Factor out 2y from both terms in the numerator:
(3y)(2y(y - 2))/(4y - 8) * (2y² - 4y)/(9y)
Factor out 2 from the denominator in the first term:
(3y)(2y(y - 2))/(2(2y - 4)) * (2y² - 4y)/(9y)
Simplify further:
(3y)(2y(y - 2))/(2(2(y - 2))) * (2y² - 4y)/(9y)
Cancel out the common factors:
(3y)(y)/(2) * (2y² - 4y)/(9y)
Simplify:
3y² * (2y² - 4y)/(18y)
Combine the like terms:
(6y^4 - 12y³)/(18y)
Simplify further by dividing each term by 6y:
6y^3 * (y - 2)/(18y)
Reduce the fraction by dividing numerator and denominator by 6:
y^3 * (y - 2)/(3y)
Therefore, the simplified expression is y^3 * (y - 2)/(3y).
4. Simplify the expression (2x³ - x² - 13x - 6) ÷ (x - 3):
To divide polynomials, we can use either long division or synthetic division. Let's use long division for this example. Set up the division as follows:
_____________________
x - 3 | 2x³ - x² - 13x - 6
Divide the first term of the dividend by the first term of the divisor:
_____________________
x - 3 | 2x³ - x² - 13x - 6
- (2x³ - 6x²)
Subtract the result from the previous step from the dividend:
_____________________
x - 3 | 2x³ - x² - 13x - 6
- (2x³ - 6x²)
________________
5x² - 13x
Bring down the next term:
_____________________
x - 3 | 2x³ - x² - 13x - 6
- (2x³ - 6x²)
________________
5x² - 13x
- (5x² - 15x)
Continue this process until all terms have been divided:
_____________________
x - 3 | 2x³ - x² - 13x - 6
- (2x³ - 6x²)
________________
5x² - 13x
- (5x² - 15x)
________________
2x - 6
The resulting quotient is 2x - 6.
Therefore, the simplified expression is 2x - 6.