Which is the root of the equation x^4-9x^3-6x^2+2=?
Answers
Step-by-step explanation:
STEP1:Equation at the end of step 1 (((x4)-(9•(x3)))+(2•3x2))+2 STEP 2 :Equation at the end of step2: (((x4) - 32x3) + (2•3x2)) + 2 STEP3:Checking for a perfect cube
3.1 x4-9x3+6x2+2 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: x4-9x3+6x2+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 6x2+2
Group 2: -9x3+x4
Pull out from each group separately :
Group 1: (3x2+1) • (2)
Group 2: (x-9) • (x3)
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 :
3.3 Find roots (zeroes) of : F(x) = x4-9x3+6x2+2
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 2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 18.00 -2 1 -2.00 114.00 1 1 1.00 0.00 x-1 2 1 2.00 -30.00
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
x4-9x3+6x2+2
can be divided with x-1
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : x4-9x3+6x2+2
("Dividend")
By : x-1 ("Divisor")
dividend x4 - 9x3 + 6x2 + 2 - divisor * x3 x4 - x3 remainder - 8x3 + 6x2 + 2 - divisor * -8x2 - 8x3 + 8x2 remainder - 2x2 + 2 - divisor * -2x1 - 2x2 + 2x remainder - 2x + 2 - divisor * -2x0 - 2x + 2 remainder 0
Quotient : x3-8x2-2x-2 Remainder: 0
Polynomial Roots Calculator :
3.5 Find roots (zeroes) of : F(x) = x3-8x2-2x-2
See theory in step 3.3
In this case, the Leading Coefficient is 1 and the Trailing Constant is -2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -9.00 -2 1 -2.00 -38.00 1 1 1.00 -11.00 2 1 2.00 -30.00
Polynomial Roots Calculator found no rational roots
please mark as brilliant it is from by book of maths