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Salve ea ch the roots. ezuation and fad all X4 -2x3_2x°+8X+ l2=o

Salve ea ch the roots. ezuation and fad all X4 -2x3_2x°+8X+ l2=o

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**Solve each equation and find all the roots.** \[ x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 \] This task involves solving the quartic polynomial equation. A quartic polynomial is a polynomial of degree four, which typically has the form: \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] In this equation, \( a = 1 \), \( b = -2 \), \( c = -7 \), \( d = 8 \), and \( e = 12 \). To find the roots, one might employ various methods such as: 1. **Factoring the Polynomial:** This involves rewriting the polynomial as a product of two or more polynomials of lower degrees, if possible. 2. **Synthetic Division or Polynomial Division:** This can be used if a potential root is known or suspected. 3. **Graphical Methods:** Plotting the polynomial graph to visually estimate the roots. 4. **Numerical Methods:** Applying algorithms like the Newton-Raphson method for approximation. 5. **Using the Rational Root Theorem:** This helps in identifying possible rational roots. The solutions to this equation (roots) can be real or complex numbers, depending on the nature of the polynomial discriminant. Identifying and working through each method will help determine all the potential roots of the equation.
Transcribed Image Text:**Solve each equation and find all the roots.** \[ x^4 - 2x^3 - 7x^2 + 8x + 12 = 0 \] This task involves solving the quartic polynomial equation. A quartic polynomial is a polynomial of degree four, which typically has the form: \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] In this equation, \( a = 1 \), \( b = -2 \), \( c = -7 \), \( d = 8 \), and \( e = 12 \). To find the roots, one might employ various methods such as: 1. **Factoring the Polynomial:** This involves rewriting the polynomial as a product of two or more polynomials of lower degrees, if possible. 2. **Synthetic Division or Polynomial Division:** This can be used if a potential root is known or suspected. 3. **Graphical Methods:** Plotting the polynomial graph to visually estimate the roots. 4. **Numerical Methods:** Applying algorithms like the Newton-Raphson method for approximation. 5. **Using the Rational Root Theorem:** This helps in identifying possible rational roots. The solutions to this equation (roots) can be real or complex numbers, depending on the nature of the polynomial discriminant. Identifying and working through each method will help determine all the potential roots of the equation.
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