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

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

문제 44

4장, 문제 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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주요 개념

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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).

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.

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.