step 1 :
2
simplify ββ
v2
equation at the end of step 1 :
2
β’ (v2)) - - 5v) + 4
v2
step 2 :
equation at the end of step 2 :
2
((2v2 - - 5v) + 4
v2
step 3 :
rewriting the whole as an equivalent fraction :
3.1 subtracting a fraction from a whole
rewrite the whole as a fraction using v2 as the denominator :
2v2 2v2 β’ v2
2v2 = =
1 v2
equivalent fraction : the fraction thus generated looks different but has the same value as the whole
common denominator : the equivalent fraction and the other fraction involved in the calculation share the same denominator
adding fractions that have a common denominator :
3.2 adding up the two equivalent fractions
add the two equivalent fractions which now have a common denominator
combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
2v2 β’ v2 - (2) 2v4 - 2
=
v2 v2
equation at the end of step 3 :
(2v4 - 2)
- 5v) + 4
v2
step 4 :
rewriting the whole as an equivalent fraction :
4.1 subtracting a whole from a fraction
rewrite the whole as a fraction using v2 as the denominator :
5v 5v β’ v2
5v = ββ =
1 v2
step 5 :
pulling out like terms :
5.1 pull out like factors :
2v4 - 2 = 2 β’ (v4 - 1)
trying to factor as a difference of squares :
5.2 factoring: v4 - 1
theory : a difference of two perfect squares, a2 - b2 can be factored into (a+b) β’ (a-b)
proof : (a+b) β’ (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 : 1 is the square of 1
check : v4 is the square of v2
factorization is : (v2 + 1) β’ (v2 - 1)
polynomial roots calculator :
5.3 find roots (zeroes) of : f(v) = v2 + 1
polynomial roots calculator is a set of methods aimed at finding values of v for which f(v)=0
rational roots test is one of the above mentioned tools. it would only find rational roots that is numbers v which can be expressed as the quotient of two integers
the rational root theorem states that if a polynomial zeroes for a rational number p/q then p is a factor of the trailing constant and q is a factor of the leading coefficient
in this case, the leading coefficient is 1 and the trailing constant is 1.
the factor(s) are:
of the leading coefficient : 1
of the trailing constant : 1
let us test
p q p/q f(p/q) divisor
-1 1 -1.00 2.00
1 1 1.00 2.00
polynomial roots calculator found no rational roots
trying to factor as a difference of squares :
5.4 factoring: v2 - 1
check : 1 is the square of 1
check : v2 is the square of v1
factorization is : (v + 1) β’ (v - 1)
adding fractions that have a common denominator :
5.5 adding up the two equivalent fractions
2 β’ (v2+1) β’ (v+1) β’ (v-1) - (5v β’ v2) 2v4 - 5v3 - 2
=
v2 v2
equation at the end of step 5 :
(2v4 - 5v3 - 2)
+ 4
v2
step 6 :
rewriting the whole as an equivalent fraction :
6.1 adding a whole to a fraction
rewrite the whole as a fraction using v2 as the denominator :
4 4 β’ v2
4 = β =
1 v2
polynomial roots calculator :
6.2 find roots (zeroes) of : f(v) = 2v4 - 5v3 - 2
see theory in step 5.3
in this case, the leading coefficient is 2 and the trailing constant is -2.
the factor(s) are:
of the leading coefficient : 1,2
of the trailing constant : 1 ,2
let us test
p q p/q f(p/q) divisor
-1 1 -1.00 5.00
-1 2 -0.50 -1.25
-2 1 -2.00 70.00
1 1 1.00 -5.00
1 2 0.50 -2.50
2 1 2.00 -10.00
polynomial roots calculator found no rational roots
adding fractions that have a common denominator :
6.3 adding up the two equivalent fractions
(2v4-5v3-2) + 4 β’ v2 2v4 - 5v3 + 4v2 - 2
=
v2 v2
checking for a perfect cube :
6.4 2v4 - 5v3 + 4v2 - 2 is not a perfect cube
trying to factor by pulling out :
6.5 factoring: 2v4 - 5v3 + 4v2 - 2
thoughtfully split the expression at hand into groups, each group having two terms :
group 1: 4v2 - 2
group 2: 2v4 - 5v3
pull out from each group separately :
group 1: (2v2 - 1) β’ (2)
group 2: (2v - 5) β’ (v3)
bad news factoring by pulling out fails :
the groups have no common factor and can not be added up to form a multiplication.
polynomial roots calculator :
6.6 find roots (zeroes) of : f(v) = 2v4 - 5v3 + 4v2 - 2
see theory in step 5.3
in this case, the leading coefficient is 2 and the trailing constant is -2.
the factor(s) are:
of the leading coefficient : 1,2
of the trailing constant : 1 ,2
let us test
p q p/q f(p/q) divisor
-1 1 -1.00 9.00
-1 2 -0.50 -0.25
-2 1 -2.00 86.00
1 1 1.00 -1.00
1 2 0.50 -1.50
2 1 2.00 6.00
polynomial roots calculator found no rational roots
final result :
2v4 - 5v3 + 4v2 - 2
v2
hope this
give me brainliest
