Question

solving polynomial equations using technology
identifying a function with zeroes that are equal to the roots of a polynomia
from least to greatest, what are the roots of the polynomial equation $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$?
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Explanation:

Step1: Rearrange the equation

First, we need to get all terms on one side of the equation to form a standard polynomial equation.
Starting with \(2x^{3}+4x^{2}-x + 5=-3x^{2}+4x + 9\), we add \(3x^{2}-4x - 9\) to both sides:
\(2x^{3}+4x^{2}+3x^{2}-x-4x + 5 - 9 = 0\)
Simplifying the like terms, we get \(2x^{3}+7x^{2}-5x - 4 = 0\)

Step2: Use technology (or rational root theorem and testing) to find roots

We can use a graphing calculator or a polynomial solver to find the roots of \(2x^{3}+7x^{2}-5x - 4 = 0\)
By testing possible rational roots (using Rational Root Theorem: possible roots are \(\pm1,\pm2,\pm4,\pm\frac{1}{2}\))

• Testing \(x = - 4\): \(2(-4)^{3}+7(-4)^{2}-5(-4)-4=2(-64)+7(16)+20 - 4=-128 + 112+20 - 4 = 0\). So \(x=-4\) is a root.
• Now we can factor the polynomial as \((x + 4)(2x^{2}-x - 1)=0\)
• Factor the quadratic \(2x^{2}-x - 1\): We need two numbers \(a\) and \(b\) such that \(a\times b=2\times(-1)=-2\) and \(a + b=-1\). The numbers are \(-2\) and \(1\). So \(2x^{2}-x - 1=2x^{2}-2x+x - 1 = 2x(x - 1)+1(x - 1)=(2x + 1)(x - 1)\)
• So the polynomial factors as \((x + 4)(2x + 1)(x - 1)=0\)
• Setting each factor equal to zero:
• \(x+4 = 0\Rightarrow x=-4\)
• \(2x + 1=0\Rightarrow x=-\frac{1}{2}\)
• \(x - 1=0\Rightarrow x = 1\)

Answer:

\(-4\), \(-\frac{1}{2}\), \(1\)