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Chapter 2: Problem 102

Find all real zeros of the function. $$g(x)=3 x^{3}-2 x^{2}+15 x-10$$

Short Answer

Expert verified
The real zeros of the function \( g(x) = 3x^3 - 2x^2 + 15x - 10 \) is x = 1. The other two zeros are non-real, given by x = 5/6 卤 \(sqrt{-95}/6i\).

Step by step solution

01

Set the Cubic Equation to Zero

Given a function \( g(x) = 3x^3 - 2x^2 + 15x - 10 \), to find the zeros, the function must be equated to zero. So, the equation to solve will be \( 3x^3 - 2x^2 + 15x - 10 = 0 \).
02

Guessing and Checking for Potential Solutions Using Synthetic Division

This step involves trying out potential solutions to the equation, mostly by evaluating integer factors of the constant term, in this case, factoring -10: \(-1, 1, -2, 2, -5, 5, -10, 10\). Using synthetic division with these potential solutions, we find that x = 1 is a solution as it results in a remainder of 0.
03

Further Solving the Resultant Quadratic Equation

After synthetic division with x = 1, a quadratic equation \( 3x^2 - 5x + 10 = 0 \) is derived. This equation can be solved using the quadratic formula: \( x = [-b 卤 sqrt(b^2 - 4ac)] / 2a \). Here a = 3, b = -5 and c = 10. After calculations, two non-real solutions are found as x = 5/6 卤 \(\sqrt{-95}/6i\).
04

Checking Found Solutions

The found solutions can be tested by substituting these back into the original equation to check if it evaluates to zero.

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Most popular questions from this chapter

Write the polynomial as the product of linear factors and list all the zeros of the function. $$h(x)=x^{3}+9 x^{2}+27 x+35$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{4}+10 x^{2}+9$$

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{4}+3 x^{3}-5 x^{2}-21 x+22\) Zero \(-3+\sqrt{2} i\)

Think About It Let \(y=f(x)\) be a cubic polynomial with leading coefficient \(a=-1\) and \(f(2)=f(i)=0\) Write an equation for \(f\)

(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. $$x^{2}+b x+4=0$$
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