Answer:
STEP
1
:
Equation at the end of step 1
((16 • (t4)) - 23t2) + 1
STEP
2
:
Equation at the end of step
2
:
(24t4 - 23t2) + 1
STEP
3
:
Trying to factor by splitting the middle term
3.1 Factoring 16t4-8t2+1
The first term is, 16t4 its coefficient is 16 .
The middle term is, -8t2 its coefficient is -8 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 16 • 1 = 16
Step-2 : Find two factors of 16 whose sum equals the coefficient of the middle term, which is -8 .
-16 + -1 = -17
-8 + -2 = -10
-4 + -4 = -8 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -4
16t4 - 4t2 - 4t2 - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
4t2 • (4t2-1)
Add up the last 2 terms, pulling out common factors :
1 • (4t2-1)
Step-5 : Add up the four terms of step 4 :
(4t2-1) • (4t2-1)
Which is the desired factorization
Trying to factor as a Difference of Squares:
3.2 Factoring: 4t2-1
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 : 4 is the square of 2
Check : 1 is the square of 1
Check : t2 is the square of t1
Factorization is : (2t + 1) • (2t - 1)
Trying to factor as a Difference of Squares:
3.3 Factoring: 4t2 - 1
Check : 4 is the square of 2
Check : 1 is the square of 1
Check : t2 is the square of t1
Factorization is : (2t + 1) • (2t - 1)
Multiplying Exponential Expressions:
3.4 Multiply (2t + 1) by (2t + 1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (2t+1) and the exponents are :
1 , as (2t+1) is the same number as (2t+1)1
and 1 , as (2t+1) is the same number as (2t+1)1
The product is therefore, (2t+1)(1+1) = (2t+1)2
Multiplying Exponential Expressions:
3.5 Multiply (2t-1) by (2t-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (2t-1) and the exponents are :
1 , as (2t-1) is the same number as (2t-1)1
and 1 , as (2t-1) is the same number as (2t-1)1
The product is therefore, (2t-1)(1+1) = (2t-1)2
Final result :
(2t + 1)2 • (2t - 1)2
Step-by-step explanation:
