Find all the other zeroes of the polynomial x4 x3 9x2 3x 18 if two of its zeroes are 3 and 3

Table of Contents Show

Step by step solution :
Step 1 :
Step 2 :
Polynomial Roots Calculator :
Polynomial Long Division :
Polynomial Roots Calculator :
Polynomial Long Division :
Step 3 :
Four solutions were found :

find other zeroes of polynomial x4+x3-9x2-3x+18 if it is given that two of its zeroes are root3 and -root3

A very good question. Keep it up...

thankyou so so so much abhinav

Check out the new look and enjoy easier access to your favorite features

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)-(x3))-32x2)+3x)+18 = 0

Step 2 :

Polynomial Roots Calculator :

2.1 Find roots (zeroes) of : F(x) = x4-x3-9x2+3x+18
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 The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : Let us test ....

P	Q	P/Q	F(P/Q)	Divisor
-1	1	-1.00	8.00
-2	1	-2.00	0.00	x+2
-3	1	-3.00	36.00
-6	1	-6.00	1188.00
-9	1	-9.00	6552.00
-18	1	-18.00	107856.00
1	1	1.00	12.00
2	1	2.00	-4.00
3	1	3.00	0.00	x-3
6	1	6.00	792.00
9	1	9.00	5148.00
18	1	18.00	96300.00

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-x3-9x2+3x+18

can be divided by 2 different polynomials,including by x-3

Polynomial Long Division :

2.2 Polynomial Long Division
Dividing :

x4-x3-9x2+3x+18 ("Dividend")

By : x-3 ("Divisor")

dividend		x4	-	x3	-	9x2	+	3x	+	18
- divisor	* x3	x4	-	3x3
remainder				2x3	-	9x2	+	3x	+	18
- divisor	* 2x2			2x3	-	6x2
remainder					-	3x2	+	3x	+	18
- divisor	* -3x1				-	3x2	+	9x
remainder							-	6x	+	18
- divisor	* -6x0						-	6x	+	18
remainder										0

Quotient : x3+2x2-3x-6 Remainder: 0

Polynomial Roots Calculator :

2.3 Find roots (zeroes) of : F(x) = x3+2x2-3x-6

See theory in step 2.1

In this case, the Leading Coefficient is 1 and the Trailing Constant is The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : Let us test ....

P	Q	P/Q	F(P/Q)	Divisor
-1	1	-1.00	-2.00
-2	1	-2.00	0.00	x+2
-3	1	-3.00	-6.00
-6	1	-6.00	-132.00
1	1	1.00	-6.00
2	1	2.00	4.00
3	1	3.00	30.00
6	1	6.00	264.00

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

x3+2x2-3x-6

can be divided with x+2

Polynomial Long Division :

2.4 Polynomial Long Division
Dividing : By : x+2 ("Divisor")

dividend		x3	+	2x2	-	3x	-	6
- divisor	* x2	x3	+	2x2
remainder					-	3x	-	6
- divisor	* 0x1
remainder					-	3x	-	6
- divisor	* -3x0				-	3x	-	6
remainder								0

Quotient : x2-3 Remainder: 0

Trying to factor as a Difference of Squares :

2.5 Factoring: x2-3

Theory : A difference of two perfect squares, A2 - B2 can be factored into Proof :

  (A+B) • (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 : 3 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step 2 :

(x2 - 3) • (x + 2) • (x - 3) = 0

Step 3 :

Theory - Roots of a product :

3.1 A product of several terms equals zero.When a product of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

3.2      Solve  :    x2-3 = 0Add 3 to both sides of the equation :
                      x2 = 3

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
                      x = ± √ 3 The equation has two real solutions
These solutions are x = ± √3 = ± 1.7321