Question

Divide and fully simplify, show work please I am stressed

Answer 1

STEP1: Equation at the end of step 1

`\frac{( {5}^(2) {x}^(2) - 4) }{( {x}^(2) - 9) } / (5x - 2) / (x + 3) \\`

STEP2:

Simplify —
`\frac{25x {}^(2) - 4}{x {}^(2) - 9 } \\`

Factoring: 25x2 - 4

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A² - AB + BA - B² =

A² - AB + AB - B² =

A² - B²

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 25 is the square of 5

Check : 4 is the square of 2

Check : x2 is the square of x1

Factorization is : (5x + 2) • (5x - 2)

Trying to factor as a Difference of Squares:

Factoring: x2 - 9

Check : 9 is the square of 3

Check : x2 is the square of x1

Factorization is : (x + 3) • (x - 3)

Polynomial Long Division :

2.3 Polynomial Long Division

Dividing : 5x + 2

("Dividend")

By : x + 3 ("Divisor")

dividend 5x + 2

- divisor * 5x⁰ 5x + 15

remainder - 13

Quotient : 5

Remainder : -13

Equation at the end of step2:

`( (5x + 2) • (5x - 2))/((x + 3) • (x - 3)) ÷ (5x - 2) ÷ (x + 3) \\`

STEP3:

Divide
`((5x+2)•(5x-2))/((x+3)•(x-3)) \: by \: 5x-2 \\`

Canceling Out :

3.1 Cancel out (5x - 2) which appears on both sides of the fraction line.

Equation at the end of step3:

`((5x + 2))/((x + 3) • (x - 3)) ÷ (x + 3) \\`

STEP4:

Divide
`(5x+2)/((x+3)•(x-3)) \: by \: x + 3 \\`

Multiplying Exponential Expressions:

4.1 Multiply (x + 3) by (x + 3)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (x+3) and the exponents are :

1 , as (x+3) is the same number as (x+3)1

and 1 , as (x+3) is the same number as (x+3)1

The product is therefore, (x+3)¹ = (x+3)²

Final result :

`\frac{5x + 2}{(x + 3) {}^(2) • (x - 3)} \\`