solve the equation 3x^3-4x^2+x+88=0 one root being (2+root7i)
Answers
Answer:
3 result(s) found
x=−
3
8
=−2.667
x=
2
4−
−28
=2−i
7
=2.0000−2.6458i
x=
2
4+
−28
=2+i
7
=2.0000+2.6458i
Explanation
Step by Step Solution:
More Icon
Step by step solution :
STEP
1
:
Equation at the end of step 1
(((3 • (x3)) - 22x2) + x) + 88 = 0
STEP
2
:
Equation at the end of step
2
:
((3x3 - 22x2) + x) + 88 = 0
STEP
3
:
Checking for a perfect cube
3.1 3x3-4x2+x+88 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 3x3-4x2+x+88
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x+88
Group 2: -4x2+3x3
Pull out from each group separately :
Group 1: (x+88) • (1)
Group 2: (3x-4) • (x2)
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) = 3x3-4x2+x+88
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 3 and the Trailing Constant is 88.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4 ,8 ,11 ,22 ,44 ,88
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 80.00
-1 3 -0.33 87.11
-2 1 -2.00 46.00
-2 3 -0.67 84.67
-4 1 -4.00 -172.00
-4 3 -1.33 72.44
-8 1 -8.00 -1712.00
-8 3 -2.67 0.00 3x+8
-11 1 -11.00 -4400.00
-11 3 -3.67 -117.33
-22 1 -22.00 -33814.00
-22 3 -7.33 -1317.56
-44 1 -44.00 -263252.00
-44 3 -14.67 -10252.00
-88 1 -88.00 -2075392.00
-88 3 -29.33 -79102.22
1 1 1.00 88.00
1 3 0.33 88.00
2 1 2.00 98.00
2 3 0.67 87.78
4 1 4.00 220.00
4 3 1.33 89.33
8 1 8.00 1376.00
8 3 2.67 119.11
11 1 11.00 3608.00
11 3 3.67 185.78
22 1 22.00 30118.00
22 3 7.33 1063.33
44 1 44.00 247940.00
44 3 14.67 8707.11
88 1 88.00 2013616.00
88 3 29.33 72394.67
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
3x3-4x2+x+88
can be divided with 3x+8
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 3x3-4x2+x+88
("Dividend")
By : 3x+8 ("Divisor")
dividend 3x3 - 4x2 + x + 88
- divisor * x2 3x3 + 8x2
remainder - 12x2 + x + 88
- divisor * -4x1 - 12x2 - 32x
remainder 33x + 88
- divisor * 11x0 33x + 88
remainder 0
Quotient : x2-4x+11 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring x2-4x+11
The first term is, x2 its coefficient is 1 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is +11
Step-1 : Multiply the coefficient of the first term by the constant 1 • 11 = 11
Step-2 : Find two factors of 11 whose sum equals the coefficient of the middle term, which is -4 .
-11 + -1 = -12
-1 + -11 = -12
1 + 11 = 12
11 + 1 = 12
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
3
:
(x2 - 4x + 11) • (3x + 8) = 0
STEP
4
:
Theory - Roots of a product
4.1 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 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Parabola, Finding the Vertex:
4.2 Find the Vertex of y = x2-4x+11
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 2.0000
Plugging into the parabola formula 2.0000 for x we can calculate the y -coordinate :
y = 1.0 * 2.00 * 2.00 - 4.0 * 2.00 + 11.0
or y = 7.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2-4x+11
Axis of Symmetry (dashed) {x}={ 2.00}
Vertex at {x,y} = { 2.00, 7.00}
Function has no real roots
I hope you understand my answer
Answer:
-8/3
Step-by-step explanation:
Concept= Quadratic Equation
Given= Polynomial with a root
To find= the value of other root
Explanation=
We have been given to solve the equation 3x³-4x²+x+88=0 with its one root being (2+√7i)
Let f(x) = 3x³-4x²+x+88=0
If one root is (2+√7i) therefore the other is (2-√7i)
so {x - (2+√7i)}{x - (2-√7i)} is a factor of f(x)
therefore,
{(x-2) -√7i}{(x-2) +√7i}
=> (x-2)² - (√7i)²
=> x²+4-4x +7 = x²-4x +11 is a factor of f(x).
Therefore dividing f(x) with x²-4x +11
we get,
f(x) = (x²-4x +11)(3x +8)
now,
3x+8=0
3x=-8
x=-8/3
Therefore the other root of the given equation is -8/3.
#SPJ3