The Polynomial Equation $x^3-4x^2+2x+10=x^2-5x-3$ Has Complex Roots $3 \pm 2i$. What Is The Other Root? Use A Graphing Calculator And A System Of Equations.A. $-3$ B. $-1$ C. $3$ D. $10$
Introduction
Solving polynomial equations can be a challenging task, especially when they involve complex roots. In this article, we will explore a method to find the other root of the given polynomial equation using a graphing calculator and a system of equations.
Understanding the Problem
The given polynomial equation is $x3-4x2+2x+10=x^2-5x-3$. We are told that the complex roots of this equation are $3 \pm 2i$. Our goal is to find the other root of this equation.
Using a Graphing Calculator
One way to approach this problem is to use a graphing calculator to visualize the graph of the polynomial equation. By graphing the equation, we can identify the x-intercepts, which correspond to the roots of the equation.
Graphing the Equation
To graph the equation, we can use a graphing calculator such as the TI-84 or TI-Nspire. First, we need to enter the equation into the calculator. We can do this by pressing the "Y=" button and entering the equation.
y = x^3 - 4x^2 + 2x + 10 - (x^2 - 5x - 3)
Next, we need to set the window settings to ensure that the graph is visible. We can do this by pressing the "WINDOW" button and adjusting the x and y limits.
Xmin = -10
Xmax = 10
Ymin = -10
Ymax = 10
Once we have set the window settings, we can graph the equation by pressing the "GRAPH" button.
Identifying the Roots
By graphing the equation, we can identify the x-intercepts, which correspond to the roots of the equation. In this case, we can see that the graph intersects the x-axis at three points: $x = -3$, $x = 3$, and $x = -1$.
Using a System of Equations
Another way to approach this problem is to use a system of equations to find the other root. We can do this by rewriting the given equation as a system of two equations.
x^3 - 4x^2 + 2x + 10 = 0
x^2 - 5x - 3 = 0
We can then solve this system of equations using substitution or elimination.
Solving the System of Equations
To solve the system of equations, we can use substitution. We can solve the second equation for $x^2$ and substitute this expression into the first equation.
x^2 = 5x + 3
x^3 - 4x^2 + 2x + 10 = 0
x^3 - (5x + 3)x + 2x + 10 = 0
x^3 - 5x^2 - 3x + 2x + 10 = 0
x^3 - 5x^2 - x + 10 = 0
We can then factor the resulting equation to find the roots.
Factoring the Equation
We can factor the resulting equation as follows:
(x - 3)(x^2 + 2x + 10/3) = 0
We can then set each factor equal to zero and solve for $x$.
Solving for $x$
We can solve for $x$ by setting each factor equal to zero.
x - 3 = 0
x = 3
x^2 + 2x + 10/3 = 0
x^2 + 2x + 10/3 = 0
We can then use the quadratic formula to solve for $x$.
Using the Quadratic Formula
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, we have:
a = 1
b = 2
c = 10/3
We can then plug these values into the quadratic formula to solve for $x$.
Solving for $x$
We can solve for $x$ by plugging the values into the quadratic formula.
x = (-2 ± √(2^2 - 4(1)(10/3))) / 2(1)
x = (-2 ± √(4 - 40/3)) / 2
x = (-2 ± √(12/3 - 40/3)) / 2
x = (-2 ± √(-28/3)) / 2
x = (-2 ± √(-28/3)) / 2
x = (-2 ± i√(28/3)) / 2
x = (-2 ± i√(28/3)) / 2
We can then simplify the expression to find the other root.
Simplifying the Expression
We can simplify the expression by multiplying the numerator and denominator by $\sqrt{3}$.
x = (-2 ± i√(28/3)) / 2
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# Q&A: The polynomial equation $x^3-4x^2+2x+10=x^2-5x-3$ has complex roots $3 \pm 2i$. What is the other root?
## Q: What is the given polynomial equation?
A: The given polynomial equation is $x^3-4x^2+2x+10=x^2-5x-3$.
## Q: What are the complex roots of the given polynomial equation?
A: The complex roots of the given polynomial equation are $3 \pm 2i$.
## Q: How can we find the other root of the given polynomial equation?
A: We can find the other root of the given polynomial equation using a graphing calculator and a system of equations.
## Q: What is the first step in using a graphing calculator to find the other root?
A: The first step in using a graphing calculator to find the other root is to enter the equation into the calculator.
## Q: How do we enter the equation into the calculator?
A: We can enter the equation into the calculator by pressing the "Y=" button and entering the equation.
## Q: What is the next step in using a graphing calculator to find the other root?
A: The next step in using a graphing calculator to find the other root is to set the window settings to ensure that the graph is visible.
## Q: How do we set the window settings?
A: We can set the window settings by pressing the "WINDOW" button and adjusting the x and y limits.
## Q: What are the window settings for this problem?
A: The window settings for this problem are:
```math
Xmin = -10
Xmax = 10
Ymin = -10
Ymax = 10
Q: What is the next step in using a graphing calculator to find the other root?
A: The next step in using a graphing calculator to find the other root is to graph the equation.
Q: How do we graph the equation?
A: We can graph the equation by pressing the "GRAPH" button.
Q: What can we see on the graph?
A: On the graph, we can see that the graph intersects the x-axis at three points: $x = -3$, $x = 3$, and $x = -1$.
Q: What is the next step in finding the other root?
A: The next step in finding the other root is to use a system of equations to find the other root.
Q: How do we rewrite the given equation as a system of equations?
A: We can rewrite the given equation as a system of equations by subtracting the second equation from the first equation.
Q: What is the resulting system of equations?
A: The resulting system of equations is:
x^3 - 4x^2 + 2x + 10 = 0
x^2 - 5x - 3 = 0
Q: How do we solve the system of equations?
A: We can solve the system of equations using substitution or elimination.
Q: What is the first step in solving the system of equations?
A: The first step in solving the system of equations is to solve the second equation for $x^2$.
Q: How do we solve the second equation for $x^2$?
A: We can solve the second equation for $x^2$ by adding $5x$ to both sides of the equation and then adding $3$ to both sides of the equation.
Q: What is the resulting expression for $x^2$?
A: The resulting expression for $x^2$ is:
x^2 = 5x + 3
Q: What is the next step in solving the system of equations?
A: The next step in solving the system of equations is to substitute the expression for $x^2$ into the first equation.
Q: How do we substitute the expression for $x^2$ into the first equation?
A: We can substitute the expression for $x^2$ into the first equation by replacing $x^2$ with $5x + 3$.
Q: What is the resulting equation?
A: The resulting equation is:
x^3 - 4(5x + 3) + 2x + 10 = 0
Q: What is the next step in solving the system of equations?
A: The next step in solving the system of equations is to simplify the resulting equation.
Q: How do we simplify the resulting equation?
A: We can simplify the resulting equation by distributing the $-4$ to the terms inside the parentheses.
Q: What is the resulting equation?
A: The resulting equation is:
x^3 - 20x - 12 + 2x + 10 = 0
Q: What is the next step in solving the system of equations?
A: The next step in solving the system of equations is to combine like terms.
Q: How do we combine like terms?
A: We can combine like terms by adding or subtracting the coefficients of the like terms.
Q: What is the resulting equation?
A: The resulting equation is:
x^3 - 18x - 2 = 0
Q: What is the next step in solving the system of equations?
A: The next step in solving the system of equations is to factor the resulting equation.
Q: How do we factor the resulting equation?
A: We can factor the resulting equation by finding the greatest common factor of the terms.
Q: What is the resulting factorization?
A: The resulting factorization is:
(x - 3)(x^2 + 2x + 10/3) = 0
Q: What is the next step in solving the system of equations?
A: The next step in solving the system of equations is to set each factor equal to zero and solve for $x$.
Q: How do we set each factor equal to zero and solve for $x$?
A: We can set each factor equal to zero and solve for $x$ by using the zero-product property.
Q: What are the solutions to the equation?
A: The solutions to the equation are:
x - 3 = 0
x = 3
x^2 + 2x + 10/3 = 0
Q: How do we solve the quadratic equation?
A: We can solve the quadratic equation by using the quadratic formula.
Q: What is the quadratic formula?
A: The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: What are the values of $a$, $b$, and $c$?
A: The values of $a$, $b$, and $c$ are:
a = 1
b = 2
c = 10/3
Q: How do we plug these values into the quadratic formula?
A: We can plug these values into the quadratic formula by substituting the values of $a$, $b$, and $c$ into the formula.
Q: What is the resulting expression?
A: The resulting expression is:
x = (-2 ± √(2^2 - 4(1)(10/3))) / 2(1)
Q: How do we simplify the expression?
A: We can simplify the expression by multiplying the numerator and denominator by $\sqrt{3}$.
Q: What is the resulting expression?
A: The resulting expression is:
x = (-2 ± i√(28/3)) / 2
Q: What is the other root of the given polynomial equation?
A: The other root of the given polynomial equation is $-1$.
Q: What is the final answer?
A: The final answer is $\boxed{-1}$.