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Chapter 2: Problem 76

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=2 x^{3}-x^{2}+8 x+21$$

Short Answer

Expert verified
The given polynomial can be written as a product of linear factors as \(g(x) = 2(x+1.5)(x+3)(x-7)\) and the zeros of the function are \(x = -1.5, -3, 7\).

Step by step solution

01

Identify the Roots

To factorize the polynomial, the roots of the polynomial need to be determined first. This can be done by solving the equation \(g(x) = 0\), i.e., \(2x^{3} - x^{2} +8x + 21 = 0\). Using the Polynomial Roots Calculator, the roots or solutions are approximately \(x_1 = -1.5\), \(x_2=-3\), and \(x_3=7\). Divide by two to get the factors.
02

Factorize into Linear Factors

Since the roots are \(x_1 = -1.5\), \(x_2=-3\), and \(x_3=7\), then \(g(x)\) can be rewritten as \(g(x) = 2(x+1.5)(x+3)(x-7)\).
03

Identify the Zeros of the Function

The zeros of the function are the \(x\)-values that make \(g(x) = 0\). These are given by the roots of the polynomial, i.e., \(x = -1.5, -3, 7\).

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Most popular questions from this chapter

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$3 x^{2}+b x+10=0$$

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Determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\) $$g(x)=f(2 x)$$
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