To find the roots of the polynomial \( p(x) = x^3 + 4x^2 - 7x - 10 \), we can apply the Rational Root Theorem and synthetic division, or we can use numerical algorithms or graphical methods.
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Finding potential rational roots using the Rational Root Theorem: The possible rational roots are factors of the constant term (-10) divided by the factors of the leading coefficient (1). The possible rational roots are: \[ \pm 1, \pm 2, \pm 5, \pm 10 \]
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Testing these potential roots:
- Let's test \( x = -5 \): \[ p(-5) = (-5)^3 + 4(-5)^2 - 7(-5) - 10 = -125 + 100 + 35 - 10 = 0 \] So, \( x = -5 \) is a root.
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Performing synthetic division with \( x + 5 \): We divide the polynomial \( p(x) \) by \( x + 5 \): \[ \begin{array}{r|rrrr} -5 & 1 & 4 & -7 & -10 \ & & -5 & 5 & 10 \ \hline & 1 & -1 & -2 & 0 \ \end{array} \] The quotient is \( x^2 - x - 2 \).
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Factoring the quadratic \( x^2 - x - 2 \): \[ x^2 - x - 2 = (x - 2)(x + 1) \] So we have: \[ p(x) = (x + 5)(x - 2)(x + 1) \]
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Finding all roots: The roots are: \[ x + 5 = 0 \Rightarrow x = -5, \] \[ x - 2 = 0 \Rightarrow x = 2, \] \[ x + 1 = 0 \Rightarrow x = -1. \]
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Writing the roots in ascending order: The roots are: \[ -5, -1, \text{ and } 2. \]
Thus, the final answer is: The roots are \( -5, -1, \) and \( 2 \).
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