Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
4. Polynomial Functions
Dividing Polynomials
Problem 44
Textbook Question
Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x^3−3x^2−11x+6.
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Given that \(-2\) is a zero of \(f(x) = 2x^3 - 3x^2 - 11x + 6\), use synthetic division to divide the polynomial by \(x + 2\).
Set up the synthetic division by writing \(-2\) on the left and the coefficients \(2, -3, -11, 6\) on the right.
Bring down the leading coefficient \(2\) to the bottom row.
Multiply \(-2\) by the number just written on the bottom row \(2\) and write the result \(-4\) under the next coefficient \(-3\).
Add \(-3\) and \(-4\) to get \(-7\), and continue the process until you complete the division, resulting in a quadratic equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the function f(x) = 2x^3 - 3x^2 - 11x + 6 is a cubic polynomial, which means it has a degree of three. Understanding polynomial functions is essential for analyzing their roots and behavior.
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Introduction to Polynomial Functions
Zeros of a Polynomial
A zero of a polynomial is a value of x that makes the polynomial equal to zero. For the given polynomial f(x), knowing that -2 is a zero means that f(-2) = 0. This information is crucial for factoring the polynomial and finding other roots, as it indicates that (x + 2) is a factor of f(x).
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Factoring Polynomials
Factoring polynomials involves expressing the polynomial as a product of simpler polynomials. Once a zero is identified, such as -2, synthetic division or polynomial long division can be used to divide the polynomial by (x + 2), simplifying the equation to find the remaining roots. This process is fundamental in solving polynomial equations.
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Introduction to Factoring Polynomials
Related Practice