what will be the remainder of the polynomial x4 +4x3 +6x2 +4x +4 is divided x + 2.
Answers
Answer:
Part 1:−
x−1)4x3−3x2+4x−2(4x2+x+54x3−4x2−+_____________________x2+4x x2−x−+_____________________5x−25x−5−+_____________________3Remainder.
Part 2:−
x−2)4x3−3x2+4x−2(4x2+5x+144x3−8x2−+_____________________5x2+4x5x2−10x−+________
Answer:
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 2 more similar replacement(s).
Step by step solution :
STEP
1
:
Equation at the end of step 1
((((x4)-(4•(x3)))-(2•3x2))+4x)+5 = 0
STEP
2
:
Equation at the end of step
2
:
((((x4) - 22x3) - (2•3x2)) + 4x) + 5 = 0
STEP
3
:
Polynomial Roots Calculator :
3.1 Find roots (zeroes) of : F(x) = x4-4x3-6x2+4x+5
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 1 and the Trailing Constant is 5.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,5
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 0.00 x+1
-5 1 -5.00 960.00
1 1 1.00 0.00 x-1
5 1 5.00 0.00 x-5
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
x4-4x3-6x2+4x+5
can be divided by 3 different polynomials,including by x-5
Polynomial Long Division :
3.2 Polynomial Long Division
Dividing : x4-4x3-6x2+4x+5
("Dividend")
By : x-5 ("Divisor")
dividend x4 - 4x3 - 6x2 + 4x + 5
- divisor * x3 x4 - 5x3
remainder x3 - 6x2 + 4x + 5
- divisor * x2 x3 - 5x2
remainder - x2 + 4x + 5
- divisor * -x1 - x2 + 5x
remainder - x + 5
- divisor * -x0 - x + 5
remainder 0
Quotient : x3+x2-x-1 Remainder: 0
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = x3+x2-x-1
See theory in step 3.1
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 0.00 x+1
1 1 1.00 0.00 x-1
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
x3+x2-x-1
can be divided by 2 different polynomials,including by x-1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : x3+x2-x-1
("Dividend")
By : x-1 ("Divisor")
dividend x3 + x2 - x - 1
- divisor * x2 x3 - x2
remainder 2x2 - x - 1
- divisor * 2x1 2x2 - 2x
remainder x - 1
- divisor * x0 x - 1
remainder 0
Quotient : x2+2x+1 Remainder: 0
Trying to factor by splitting the middle term
Step-by-step explanation:
the above answer is correct