Roots of polynomial equations 4

In summary, the product of two of the roots of the equation is equal to the product of the other two roots. Using Vieta's formulas, we can prove that a*d^2 = b^2 * e.
  • #1
Erfan1
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The product of two of the roots of the equation ax^4 + bx^3 + cx^2 + dx + e = 0 is equal to the product of the other two roots. Prove that a*d^2 = b^2 * e
 
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  • #2
Erfan said:
The product of two of the roots of the equation ax^4 + bx^3 + cx^2 + dx + e = 0 is equal to the product of the other two roots. Prove that a*d^2 = b^2 * e

If we let $p, q, r$ and $s$ to represents the 4 roots of the equation $ax^4 + bx^3 + cx^2 + dx + e = 0$, we're told that $pq=rs$.

We can also apply Vieta's formulas to get the following relations between a, b, e and d:

$p+q+r+s=-\frac{b}{a}$---(1)

$pqr+pqs+prs+qrs=-\frac{d}{a}$---(2)

$pqrs=\frac{e}{a}$---(3)Simplifying (2) yields

$pq(r+s)+rs(p+q)=-\frac{d}{a}$

$rs(r+s)+rs(p+q)=-\frac{d}{a}$ since $pq=rs$

$rs(r+s+p+q)=-\frac{d}{a}$

$rs(-\frac{b}{a})=-\frac{d}{a}$ since $p+q+r+s=-\frac{b}{a}$---(1)

$rs=-\frac{d}{b}$

Substituting $pq=rs$ and $rs=-\frac{d}{b}$ into equation (3) gives

$pqrs=\frac{e}{a}$

$(rs)rs=\frac{e}{a}\rightarrow\;\;(rs)^2=\frac{e}{a}$.

Hence, $(-\frac{d}{b})^2=\frac{e}{a}$ and this gives $ad^2=eb^2$. So we're done.
 

FAQ: Roots of polynomial equations 4

What are polynomial equations?

Polynomial equations are mathematical expressions that consist of variables and coefficients, connected by operations of addition, subtraction, and multiplication. They can be written in the form of ax^n + bx^(n-1) + ... + k = 0, where a, b, and k are constants and n is a non-negative integer representing the degree of the polynomial.

What are the roots of a polynomial equation?

The roots of a polynomial equation are the values of the variable that make the equation equal to zero. They can be real or complex numbers, and the number of roots is equal to the degree of the polynomial.

How do you solve polynomial equations?

To solve polynomial equations, you can use various methods such as factoring, the quadratic formula, or synthetic division. You can also use numerical methods like Newton's method or the bisection method to approximate the roots of a polynomial equation.

What is the relationship between the roots of a polynomial equation and its graph?

The roots of a polynomial equation are the x-intercepts of its graph. This means that if you plot the equation on a graph, the points where the graph intersects the x-axis are the roots of the polynomial.

What is the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra states that every polynomial equation of degree n has n complex roots, including repeated roots. This means that a polynomial equation of degree 4 can have up to 4 complex roots.

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