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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 44
Capitolo 4, Problema 44

Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x3−3x2−11x+6.

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Since -2 is a zero of the polynomial function \(f(x) = 2x^{3} - 3x^{2} - 11x + 6\), use synthetic division or polynomial long division to divide \(f(x)\) by the factor corresponding to this zero, which is \((x + 2)\).
Set up the synthetic division by writing the coefficients of \(f(x)\): 2, -3, -11, and 6. Then perform the division using -2 as the divisor.
After completing the synthetic division, write down the quotient polynomial, which will be a quadratic expression of the form $ax^{2} + bx + c$.
Solve the quadratic equation obtained from the quotient by using factoring, completing the square, or the quadratic formula to find the remaining zeros of \(f(x)\).
Combine the zero \(x = -2\) with the solutions from the quadratic to write the complete solution set for the equation \(2x^{3} - 3x^{2} - 11x + 6 = 0\).

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Polynomial Zeros and Roots

A zero or root of a polynomial is a value of x that makes the polynomial equal to zero. Knowing a zero helps factor the polynomial and find other roots. For example, if -2 is a zero of f(x), then (x + 2) is a factor of f(x).
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Imaginary Roots with the Square Root Property

Polynomial Division (Synthetic or Long Division)

Polynomial division is used to divide a polynomial by a binomial factor, simplifying the polynomial. Synthetic division is a shortcut method when dividing by linear factors like (x - c). Dividing by (x + 2) will reduce the cubic to a quadratic.
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Introduction to Factoring Polynomials

Factoring Quadratic Polynomials

After division, the resulting quadratic can be factored to find the remaining zeros. Factoring involves expressing the quadratic as a product of two binomials. If factoring is difficult, the quadratic formula can be used to find the roots.
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Introduction to Factoring Polynomials
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