Exercise 145 Page 300

Practice makes perfect

Notice that the fractions have different denominators. This means we cannot add them yet. Before we make any changes to give them a common denominator, we will factor the denominator in the first fraction.

Factor the Denominator

Again, we will use a generic rectangle and a diamond problem. We know that x^2 and 6 go into the lower left and upper right corner of the generic rectangle.

To fill in the remaining two corners, we need to find two x-terms that sum to 5x and have a product of 6x^2.

Notice that both the product and the sum are positive. This means both factors must be positive. |c|c|c|c|c| [-0.8em] Product & ax(bx) & ax+bx & Sum & 5x? [0.2em] [-0.8em] 6x^2 & x(6x) & x+6x & 7x & * [0.3em] [-0.8em] 6x^2 & 2x(3x) & 2x+3x & 5x & ✓ [0.3em] When one factor is 2x and the other is 3x we have a product of 6x^2 and a sum of 5x. Now we can complete the diamond and generic rectangle.

To factor the expression, we add each side of the generic rectangle and multiply the sums. x^2+5x+6 ⇕ (x+2)(x+3)

Continuing the Simplification

By replacing the denominator with the factored expression, we can continue simplifying the expression.

4/(x+2)(x+3)+2x/x+2

ExpandFrac

a/b=a * (x+3)/b * (x+3)

4/(x+2)(x+3)+2x(x+3)/(x+2)(x+3)

AddFrac

Add fractions

4+2x(x+3)/(x+2)(x+3)

▼

Simplify numerator

Distr

Distribute 2x

4+2x^2+6x/(x+2)(x+3)

CommutativePropAdd

Commutative Property of Addition

2x^2+6x+4/(x+2)(x+3)

SplitIntoFactors

Split into factors

2* x^2+2* 3x+2* 2/(x+2)(x+3)

FactorOut

Factor out 2

2(x^2+3x+2)/(x+2)(x+3)

To continue simplifying, we have to factor the trinomial in the numerator.

Factor the Numerator

Again, we will use a generic rectangle and a diamond problem. We know that x^2 and 2 go into the lower left and upper right corner of the generic rectangle.

To fill in the remaining two corners, we need to find two x-terms that sum to 3x and have a product of 2x^2.

Notice that both the product and the sum are positive. This means both factors must be positive. |c|c|c|c|c| [-0.8em] Product & ax(bx) & ax+bx & Sum & 3x? [0.2em] [-0.8em] 2x^2 & x(2x) & x+2x & 3x & ✓ [0.3em] When one factor is x and the other is 2x we have a product of 2x^2 and a sum of 3x. Now we can complete the diamond and generic rectangle.

To factor the expression, we add each side of the generic rectangle and multiply the sums. x^2+3x+2 ⇕ (x+1)(x+2)

Continuing the Simplification

By replacing the numerator with the factored expression, we can continue simplifying the expression.

2 (x+1)(x+2)/(x+2)(x+3)

ReduceFrac

a/b=.a /(x+2)./.b /(x+2).

2(x+1)/x+3

Examining the expression, we notice that the fractions have different denominators. Therefore, we must first give them the same denominator to continue simplifying.

3x^2+x/(2x+1)^2-3/2x+1

ExpandFrac

a/b=a * (2x+1)/b * (2x+1)

3x^2+x/(2x+1)^2-3(2x+1)/(2x+1)^2

SubFrac

Subtract fractions

3x^2+x-3(2x+1)/(2x+1)^2

Distr

Distribute - 3

3x^2+x-6x-3/(2x+1)^2

SimpTerms

Simplify terms