f
′
(x)=
3
(5x
3
−2x
2
)
7
(3x
2
−4x)
5
Answers
Answer:
5.1 Factoring: 4x5-x4-5x3+2x2+4x-4
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 2x2-5x3
Group 2: 4x5-x4
Group 3: 4x-4
Pull out from each group separately :
Group 1: (5x-2) • (-x2)
Group 2: (4x-1) • (x4)
Group 3: (x-1) • (4)
Looking for common sub-expressions :
Group 1: (5x-2) • (-x2)
Group 3: (x-1) • (4)
Group 2: (4x-1) • (x4)
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 :
5.2 Find roots (zeroes) of : F(x) = 4x5-x4-5x3+2x2+4x-4
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x 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 4 and the Trailing Constant is -4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,2 ,4
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -6.00
-1 2 -0.50 -5.06
-1 4 -0.25 -4.80
-2 1 -2.00 -108.00
-4 1 -4.00 -4020.00
1 1 1.00 0.00 x-1
1 2 0.50 -2.06
1 4 0.25 -2.95
2 1 2.00 84.00
4 1 4.00 3564.00
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
4x5-x4-5x3+2x2+4x-4
can be divided with x-1
Polynomial Long Division :
5.3 Polynomial Long Division
Dividing : 4x5-x4-5x3+2x2+4x-4
("Dividend")
By : x-1 ("Divisor")
dividend 4x5 - x4 - 5x3 + 2x2 + 4x - 4
- divisor * 4x4 4x5 - 4x4
remainder 3x4 - 5x3 + 2x2 + 4x - 4
- divisor * 3x3 3x4 - 3x3
remainder - 2x3 + 2x2 + 4x - 4
- divisor * -2x2 - 2x3 + 2x2
remainder 4x - 4
- divisor * 0x1
remainder 4x - 4
- divisor * 4x0 4x - 4
remainder 0
Quotient : 4x4+3x3-2x2+4 Remainder: 0
Polynomial Roots Calculator :
5.4 Find roots (zeroes) of : F(x) = 4x4+3x3-2x2+4
See theory in step 5.2
In this case, the Leading Coefficient is 4 and the Trailing Constant is 4.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,2 ,4
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 3.00
-1 2 -0.50 3.38
-1 4 -0.25 3.84
-2 1 -2.00 36.00
-4 1 -4.00 804.00
1 1 1.00 9.00
1 2 0.50 4.12
1 4 0.25 3.94
2 1 2.00 84.00
4 1 4.00 1188.00
Polynomial Roots Calculator found no rational roots
Final result :
(4x4 + 3x3 - 2x2 + 4) • (x - 1)
Step-by-step explanation:
HOPE It Helps you