simplify the rational expression t^2=2t-24/t^2-36 and state any restrictions on the variable
Accepted Solution
A:
Answer: t - 4
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t - 6
Step-by-step 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.