Can someone show me the steps for how to solve:
1.) f(x) = 4x^4 -3x^3 +2x^2 +x-3
2.) f(x) = x^3 - 2x^2 -5x +6
3.) f(x) = -3x^3 -2x^2 + x -1
Thank you so much :)
You only typed half the problem. Perhaps you are supposed to graph, not solve?
sorry, i forgot to say the instructions said "find all possible rational roots"
Result:
The roots of the polynomial 4x4 - 3x3 + 2x2 + x - 3 are:
X1 = 0.301 + 1.002*i
X2 = 0.301 - 1.002*i
X3 = 0.905
X4 = -0.757
That is from:
http://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php
As you can see you really have to graph that polynomial to find the roots. If you have a graphing calculator, try that
Notice that it only crosses the x axis twice and then has a complex pair for the third and fourth roots
For the second one you can see that one is a root by inspection.
Then you can divide by (x-1) to get a quadratic which you can solve.
By the way if you tried all possible rational roots results on the first problem using factors of the first and last term 3 and 4 ( 3, 2, 4) you would have found that it has no rational factors by the rational root test.
thank you so much for your help ! :)
Certainly! I'd be happy to explain how to solve these equations step by step.
1.) To solve the equation f(x) = 4x^4 - 3x^3 + 2x^2 + x - 3, we need to find the values of x that make f(x) equal to zero. These values are called the roots or solutions of the equation.
Step 1: Set f(x) equal to zero:
4x^4 - 3x^3 + 2x^2 + x - 3 = 0
Step 2: Simplify the equation if necessary:
There isn't much simplification we can do with this equation, so let's move on to the next step.
Step 3: Factor the equation, if possible:
Unfortunately, this equation cannot be factored easily. In such cases, we can use numerical methods to approximate the solutions or use a graphing calculator or software.
Step 4: Solve for x using numerical methods or a graphing calculator:
To get an approximate solution, you can use methods like the Newton-Raphson method or the bisection method. Alternatively, you can use a graphing calculator or software to find the x-intercepts of the graph of the equation, which represent the solutions.
2.) To solve the equation f(x) = x^3 - 2x^2 - 5x + 6, we follow the same steps as before.
Step 1: Set f(x) equal to zero:
x^3 - 2x^2 - 5x + 6 = 0
Step 2: Simplify the equation if necessary:
No simplification is needed here.
Step 3: Factor the equation, if possible:
This equation can be factored using techniques such as synthetic division or polynomial long division. However, it may be easier to use numerical methods or a graphing calculator.
Step 4: Solve for x using numerical methods or a graphing calculator:
Similar to the previous equation, you can use numerical methods like Newton-Raphson or the bisection method, or use a graphing calculator or software to find the x-intercepts of the graph.
3.) To solve the equation f(x) = -3x^3 - 2x^2 + x - 1, we'll repeat the same steps as before.
Step 1: Set f(x) equal to zero:
-3x^3 - 2x^2 + x - 1 = 0
Step 2: Simplify the equation if necessary:
No simplification is necessary here.
Step 3: Factor the equation, if possible:
Once again, this equation isn't easily factorable, so we proceed to the next step.
Step 4: Solve for x using numerical methods or a graphing calculator:
Use numerical methods or a graphing calculator to find the x-intercepts of the graph of the equation, which will represent the solutions.
Remember, these steps provide a general approach to solving equations. Depending on the complexity of the equation, other techniques may need to be applied.