Multiplying and Dividing Rational Expressions

Questions

1. Simplify

2. Simplify

3. Simplify

4. Simplify

5. Simplify

6. Simplify

7. Simplify

Solutions

1. Simplify Factor all polynomials :

x² + 3x − 10 = (x − 2)(x + 5) two numbers whose product is −10 sum is 3: −2, 5
x² − 3x − 4 = (x − 4)(x + 1) two numbers whose product is −4 sum is −3: −4, 1
x² + x − 20 = (x + 5)(x − 4) two numbers whose product is −20 sum is 1: −4, 5
x² + 4x + 3 = (x + 3)(x + 1) two numbers whose product is 3 sum is 4: 1, 3

Simplify polynomial multiplication .

and x − 4 ≠ 0, x + 1 ≠ 0, x + 5 ≠ 0

2. Simplify Factor all polynomials :

x² − x − 20 = (x − 5)(x + 4) two numbers whose product is −20 sum is −1: −5, 4
x² − 3x − 10 = (x − 5)(x + 2) two numbers whose product is −10 sum is −3: −5, 2
x² + 7x + 10 = (x + 5)(x + 2) two numbers whose product is 10 sum is 7: 5, 2
x² + 4x − 5 = (x + 5)(x − 1) two numbers whose product is −5 sum is 4: 5,−1

Simplify polynomial multiplication .

and x + 5 ≠ 0, x − 5≠ 0, x + 2 ≠ 0

3. Simplify Factor all polynomials:

6x² + 17x + 10
= 6x² + 12x + 5x + 10 two numbers whose product is 60 sum is 17: 12, 5
= 6x(x + 2) + 5(x + 2) Factor by grouping
= (6x + 5)(x + 2)

36x² − 25 = (6x − 5)(6x + 5) Difference of squares

Simplify polynomial division .

Simplify polynomial multiplication.

and 6x + 5 ≠ 0, 6x − 5 ≠ 0

4. Simplify Factor all polynomials:

4x² + 12x + 9
= 4x² + 6x + 6x + 9 two numbers whose product is 36 sum is 12: 6, 6
= 2x(2x + 3) + 3(2x + 3) Factor by grouping
= (2x + 3)(2x + 3) hey–this was a perfect square !

4x² − 9 = (2x + 3)(2x − 3) Difference of squares
6x − 9 = 3(2x − 3) common factor

Simplify polynomial division .

Simplify polynomial multiplication .

and 2x − 3 ≠ 0, 2x + 3 ≠ 0

5. Simplify Factor all polynomials (let the y tag along with the constants):

3x² + 12xy + 12y²
= 3x² + 6xy + 6xy + 12y² two numbers whose product is 36 sum is 12: 6, 6
= 3x(x + 2y) + 6y(x + 2y) Factor by grouping
= (3x + 6y)(x + 2y) = 3(x + 2y)(x + 2y) hey–this was a perfect square !

x² + 4xy + 3y² = (x + 1y)(x + 3y) = (x + y)(x + 3y) two numbers whose product is 3 sum is 4: 1, 3
4x + 8y = 4(x + 2y) common factor

Simplify polynomial division .

Simplify polynomial multiplication.

and x + 2y ≠ 0, x + y ≠ 0

6. Simplify Factor all polynomials:

5y² + 17y + 6
= 5y² + 15y + 2y + 6 two numbers whose product is 30 sum is 17: 15, 2
= 5y(y + 3) + 2(y + 3) Factor by grouping
= (5y + 2)(y + 3)

10y² + 9y + 2
= 10y² + 5y + 4y + 2 two numbers whose product is 20 sum is 9: 5, 4
= 5y(2y + 1) + 2(2y + 1) Factor by grouping
= (5y + 2)(2y + 1)

2y² + 5y − 3 = 2y² + 6y − 1y − 3 two numbers whose product is −6 sum is 5: 6,−1
= 2y(y + 3) − 1(y + 3) Factor by grouping
= (2y − 1)(y + 3)

4y² − 1 = (2y − 1)(2y + 1) difference of squares

Simplify polynomial multiplication .

and 5y + 2 ≠ 0, y + 3 ≠ 0, 2y + 1 ≠ 0, 2y − 1 ≠ 0

7. Simplify Factor all polynomials:

x² + 8x + 15 = (x + 5)(x + 3) two numbers whose product is 15 sum is 8: 5, 3

2x² + 11x + 5
= 2x² + 10x + 1x + 5 two numbers whose product is 10 sum is 11: 10, 1
= 2x(x + 5) + 1(x + 5) Factor by grouping
= (2x + 1)(x + 5)

2x² − 7x − 4
= 2x² − 8x + 1x − 4 two numbers whose product is −8 sum is −7: −8, 1
= 2x(x − 4) + 1(x − 4) Factor by grouping
= (2x + 1)(x − 4)

x² + 6x + 9 = (x + 3)(x + 3) two numbers whose product is 9 sum is 6: 3, 3

Simplify polynomial division .

Simplify polynomial multiplication .

and x + 5 ≠ 0, x + 3 ≠ 0, 2x + 1 ≠ 0