What is the root between -3 and -2 for the equation y^3 - 3y + 4 = 0?

In summary, one of the roots of the equation y^3 - 3y + 4 = 0 lies between -3 and -2. This can be shown by considering the function f(y) = y^3 - 3y + 4 and evaluating f(-3) and f(-2). By evaluating these values, it can be seen that the function must have been zero at some point between -3 and -2, thus proving that one of the roots lies within this interval. Additionally, by considering the function in the immediate neighborhood of a root, it can be determined that the function must cross the x-axis at the root, providing further evidence for the existence of a root between -3 and -2.
  • #1
scoutfai
70
0
show that for the equation y^3 - 3y + 4 = 0 ,
one of the root lies between -3 and -2


i don't know how to show that one of the root lies between -3 and -2, but i can show that one of the root ( or more ) is smaller than - 3^(1/2), pay in mind that -3 and -2 are also smaller than -3^(1/2).
Here is my method, but it doesn't solve the question, i wrote it just for your reference so you got more idea to solve it. U help is meaningful to me, thanks you !

y^3 - 3y + 4 = (y)(y^2 - 3) + 4 = 0
so this mean in order to make the equation becomes zero,
the term (y)( y^2 - 3) must equal -4, in order word, it must less than zero...
hence it is right to write (y)(y^2 - 3) < 0
by number line method or graph method, we know that the range of y for this inequality is y < -3^(1/2) or 0 < y < 3^(1/2)
so this imply that one of the root ( or more ) is smaller than -3^(1/2).

But, this is not the correct answer, we need to show it lies between -3 and -2 , not the negative squate root of 3 !
Please, any expert, if u know the method, please show me as soon as possible, u help is meaningful to me!
 
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  • #2
what happens when you let y=-3? and when you let y=-2? so at some point in between -3 and -2 the function must have been zero because...?
 
  • #3
when matt says let y=-3,he wants u to consider the function
f(y)=y^3 - 3y + 4
and evaluate f(-3).
(Just thought i would post this because many ppl who have asked me such question don't understand what we are trying to do here, they find the zero crossing argument quite difficult to understand)

-- AI
 
  • #4
Or, more in general, what can you say about f(x) in the immediate neighborhood, but on opposite sides of a root (ie : f(x'+a) and f(x'-a), when x' is a root of f) ?
 

FAQ: What is the root between -3 and -2 for the equation y^3 - 3y + 4 = 0?

What is a polynomial inequality?

A polynomial inequality is a mathematical statement that compares two polynomial expressions using the symbols <, >, ≤, ≥, or ≠. It represents a relationship between two quantities that are not necessarily equal.

How do you solve a polynomial inequality?

To solve a polynomial inequality, you need to first rearrange the terms so that the inequality is set equal to zero. Then, you can use algebraic techniques such as factoring, the quadratic formula, or the rational root theorem to find the roots of the polynomial. These roots will help you determine the intervals where the inequality is true.

What is the difference between a polynomial inequality and an equation?

The main difference between a polynomial inequality and an equation is that an inequality represents a relationship between two quantities that are not necessarily equal, while an equation represents a relationship between two quantities that are equal. In other words, an inequality has a range of solutions, while an equation has a specific solution.

Are there any special cases when solving a polynomial inequality?

Yes, there are a few special cases when solving a polynomial inequality. These include when the inequality involves absolute value, when the inequality involves fractions, and when the inequality involves more than one variable. In these cases, additional steps and considerations may be necessary to find the solution.

How are polynomial inequalities used in real life?

Polynomial inequalities are used in various fields of science and engineering, such as economics, physics, and computer science. They can be used to model and analyze real-world situations, such as predicting stock market trends, optimizing the efficiency of a machine, or determining the optimal path for a spacecraft. They are also used in everyday life, such as in budgeting, managing resources, and making decisions based on cost-benefit analysis.

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