Diketahui persamaan x ^ 4 - 4ax ^ 3 + (3a + b)x ^ 2 + (4a - 7b + c)x + 60 = 0. Akar-akarnya adalah x₁ = 1, X₂ = 2, dan x3 - x4 = 1. Tentukan nilai dari a, b, c, x3, dan x4

Given the polynomial equation:
\[ x^4 - 4ax^3 + (3a + b)x^2 + (4a - 7b + c)x + 60 = 0 \]
with roots \( x_1 = 1 \), \( x_2 = 2 \), and \( x_3, x_4 \) such that \( x_3 - x_4 = 1 \).

Since \( x_1 = 1 \) and \( x_2 = 2 \) are roots of the polynomial, we can factorize the polynomial as follows:
\[ (x - 1)(x - 2)(x - x_3)(x - x_4) \]

Given \( x_3 - x_4 = 1 \), let \( x_3 = k \) and \( x_4 = k - 1 \).

The polynomial can now be written as:
\[ (x - 1)(x - 2)(x - k)(x - (k - 1)) \]

We need to expand this polynomial and compare the coefficients with the given polynomial to determine \( a \), \( b \), \( c \), \( x_3 \), and \( x_4 \).

First, expand the polynomial:
\[ (x - 1)(x - 2) = x^2 - 3x + 2 \]
\[ (x - k)(x - (k - 1)) = (x - k)(x - k + 1) = x^2 - (2k - 1)x + k(k - 1) \]

Next, expand the product of these two quadratics:
\[ (x^2 - 3x + 2)(x^2 - (2k - 1)x + k(k - 1)) \]

To find the expanded form, we multiply term by term:
\[
(x^2 - 3x + 2)(x^2 - (2k - 1)x + k^2 - k) = x^4 - (2k - 1)x^3 + k^2 - k x^2 - 3x(x^2 - (2k - 1)x + k^2 - k) + 2(x^2 - (2k - 1)x + k^2 - k)
\]

Simplifying this, we get:
\[
x^4 - (2k + 1)x^3 + (k^2 - k - 3x^2 + 3(2k - 1)x - 3(k^2 - k)) + 2x^2 - 2(2k - 1)x + 2(k^2 - k)
\]

Combine like terms:
\[
x^4 - (2k + 1)x^3 + (k^2 - k + 2 - 3(k^2 - k) - 2(2k - 1) + 3(2k - 1))x^2 + (3(2k - 1) - 2(2k - 1))x + (3(k^2 - k) + 2(k^2 - k))
\]

\[ x^4 - (2k + 1)x^3 + (3a + b)x^2 + (4a - 7b + c)x + 60 = 0 \]

By comparing the coefficients of the expanded polynomial with the given polynomial, we can find the values of \( a \), \( b \), and \( c \).

To find the coefficients of \( x^2 \), \( x \), and the constant term:
\[
3a + b = 3k^2 - 7k + 6
4a - 7b + c = -2k^2 + 8k - 2
k(k-1) = 30
\]

Thus, by solving the equations:
\[
k(k - 1) = 30 \Rightarrow k^2 - k - 30 = 0 \Rightarrow (k - 6)(k + 5) = 0 \Rightarrow k = 6, k = -5
\]

By substituting k=6:
Thus,
\[
a = 4/3 
b = -22/3
c = 84
\]

Thus \( a = 4/3 \), \( b = -22/3 \), \( c = 84 \), \( x_3 = 6\), and \( x_4 = 5 \).

Would you like more details or have any questions about the solution?

Here are 8 relative questions expanding on the provided information:

1. What are the properties of polynomial roots?
2. How do you factorize a polynomial when the roots are known?
3. Can you explain the relationship between the coefficients and roots of a polynomial?
4. How can the Rational Root Theorem help in solving polynomial equations?
5. What is the significance of the Fundamental Theorem of Algebra in finding polynomial roots?
6. How do you use synthetic division to verify the roots of a polynomial?
7. Can you explain how the Vieta's formulas relate to the coefficients of a polynomial?
8. How can polynomials be used to model real-world problems?

**Tip:** When solving polynomial equations, always verify the roots by substituting them back into the original equation to ensure they satisfy the equation.

Explore how to solve a polynomial equation with specific roots x₁ = 1, x₂ = 2, and x₃ - x₄ = 1. Understand the relationship between the roots and coefficients of the polynomial using Vieta's formulas and factorization techniques. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation