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Please help!!!!

simplify the rational expression. state any restrictions on the variable.
t^2+2t-24/t^2-36

2 Answers

3 votes

Answer:

  • t⁴ + t³ - 36t - 24

Explanation:

Given :

  • t² + 2t - 24/t² - 36

Solving :

  • Multiply throughout with t² so the polynomial is easier to simplify
  • t² (t² + 2t - 24/t² - 36)
  • t⁴ + t³ - 24 - 36t²
  • t⁴ + t³ - 36t - 24
User Myo
by
8.3k points
5 votes

Answer: hope this helps! :)

t - 4

—————

t - 6

Explanation:

t2 + 2t - 24

Simplify ————————————

t2 - 36

Trying to factor by splitting the middle term

Factoring t2 + 2t - 24

The first term is, t2 its coefficient is 1 .

The middle term is, +2t its coefficient is 2 .

The last term, "the constant", is -24

Multiply the coefficient of the first term by the constant 1 • -24 = -24

Find two factors of -24 whose sum equals the coefficient of the middle term, which is 2 .

-24 + 1 = -23

-12 + 2 = -10

-8 + 3 = -5

-6 + 4 = -2

-4 + 6 = 2 That's it

Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and 6

t2 - 4t + 6t - 24

Add up the first 2 terms, pulling out like factors :

t • (t-4)

Add up the last 2 terms, pulling out common factors :

6 • (t-4)

Step-5 : Add up the four terms of step 4 :

(t+6) • (t-4)

Which is the desired factorization

Trying to factor as a Difference of Squares :

1.2 Factoring: t2-36

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 : 36 is the square of 6

Check : t2 is the square of t1

Factorization is : (t + 6) • (t - 6)

Canceling Out :

Cancel out (t + 6) which appears on both sides of the fraction line.

User Flake
by
8.5k points

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