HOW DO YOU FACTOR X^4 + X^3 + X^2 + X + 1?

x^4-x^3-x^2+1

This solution đơn hàng with finding the roots (zeroes) of polynomials.

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Step by Step Solution

Step 1 :

Checking for a perfect cube :1.1x4-x3-x2+1 is not a perfect cube

Trying to factor by pulling out :

1.2 Factoring: x4-x3-x2+1 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: -x2+1Group 2: x4-x3Pull out from each group separately :Group 1: (-x2+1) • (1) = (x2-1) • (-1)Group 2: (x-1) • (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 :

1.3 Find roots (zeroes) of : F(x) = x4-x3-x2+1Polynomial Roots Calculator is a phối of methods aimed at finding values ofxfor which F(x)=0 Rational Roots demo 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 integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then p is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 & the Trailing Constant is 1. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us chạy thử ....

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PQP/QF(P/Q)Divisor

	-1		1		-1.00		2.00
	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 phường originate from P/Q reduced to lớn its lowest terms In our case this means that x4-x3-x2+1can be divided with x-1

Polynomial Long Division :

1.4 Polynomial Long Division Dividing : x4-x3-x2+1("Dividend") By:x-1("Divisor")

dividend			x4	-	x3	-	x2			+	1
-divisor	* x3		x4	-	x3
remainder						-	x2			+	1
-divisor	* 0x2
remainder						-	x2			+	1
-divisor	* -x1					-	x2	+	x
remainder								-	x	+	1
-divisor	* -x0							-	x	+	1
remainder											0

Quotient : x3-x-1 Remainder: 0

Polynomial Roots Calculator :

1.5 Find roots (zeroes) of : F(x) = x3-x-1See theory in step 1.3 In this case, the Leading Coefficient is 1 & the Trailing Constant is -1. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 Let us chạy thử ....

PQP/QF(P/Q)Divisor

	-1		1		-1.00		-1.00
	1		1		1.00		-1.00

Polynomial Roots Calculator found no rational roots