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Chapter 2: Problem 59

Use synthetic division to show that \(x\) is a solution of the third-degree polynomial equation, and use the result to factor the polynomial completely. List all real solutions of the equation. \(x^{3}-7 x+6=0, \quad x=2\)

Short Answer

Expert verified
So, after performing synthetic division and factoring, it is confirmed that \(x = 2\) is indeed a solution, and the other solutions to the equation \(x^3 - 7x + 6 = 0\) are \(x = 1\) and \(x = -3\). Thus, the polynomial can be factored to \((x-2)(x-1)(x+3)\).

Step by step solution

01

Apply Synthetic division

According to the problem, a given solution is \(x = 2\). It will be used as the divider in synthetic division. Begin by writing the coefficients of the polynomial at the top of your synthetic division tableau. Then bring down the first coefficient (which is 1 for \(x^3\)), and then multiply it by 2 (the proposed solution). Write this result below the second coefficient and add them together. Repeat the process for all coefficients.
02

Verify if \(x = 2\) is a solution

If \(x = 2\) is a solution, the remainder should be zero at the last column of your synthetic division. Inspect your results.
03

Write Down the Factored Form of Polynomial

If indeed \(x = 2\) is a solution, the other part of the factored form of the polynomial can be obtained directly from the synthetic division. It is represented by the resulting coefficients from the synthetic division process, starting from the lower-degree term. Once you have this, use the zero-product property to set each factor equal to zero and solve for \(x\).
04

Solve the Quadratic Equation

Simplify and solve the resulting quadratic equation. You can use the quadratic formula or factoring, depending on the equation. Write down all the solutions.

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