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Question 4(Multiple Choice Worth 1 points) (07.02 MC) Factor completely x^3-2x^2-5x+10 (x-2)(x^2-5) (x+2)(x^2+5) (x-2)(x^2+5) (x+2)(x^2-5)

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Question 4(Multiple Choice Worth 1 points)
(07.02 MC)
Factor completely x^3-2x^2-5x+10
(x-2)(x^2-5)
(x+2)(x^2+5)
(x-2)(x^2+5)
(x+2)(x^2-5)

Question 4(Multiple Choice Worth 1 points) (07.02 MC) Factor completely x^3-2x^2-5x+10 (x-2)(x^2-5) (x+2)(x^2+5) (x-2)(x^2+5) (x+2)(x^2-5)

Solução

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RubenVeterano · Tutor por 10 anos

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To factor the polynomial \(x^3 - 2x^2 - 5x + 10\) completely, we need to find the roots of the polynomial and express it as a product of linear factors.<br /><br />First, let's use the Rational Root Theorem to identify possible rational roots. The Rational Root Theorem states that any rational root of the polynomial \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\) is a factor of the constant term \(a_0\) divided by a factor of the leading coefficient \(a_n\).<br /><br />For the polynomial \(x^3 - 2x^2 - 5x + 10\):<br />- The constant term \(a_0 = 10\)<br />- The leading coefficient \(a_n = 1\)<br /><br />The possible rational roots are the factors of 10: \(\pm 1, \pm 2, \pm 5, \pm 10\).<br /><br />We will test these possible roots by substituting them into the polynomial to see if they yield zero.<br /><br />Let's start with \(x = 1\):<br />\[ 1^3 - 2(1)^2 - 5(1) + 10 = 1 - 2 - 5 + 10 = 4 \neq 0 \]<br /><br />Next, \(x = -1\):<br />\[ (-1)^3 - 2(-1)^2 - 5(-1) + 10 = -1 - 2 + 5 + 10 = 12 \neq 0 \]<br /><br />Next, \(x = 2\):<br />\[ 2^3 - 2(2)^2 - 5(2) + 10 = 8 - 8 - 10 + 10 = 0 \]<br /><br />So, \(x = 2\) is a root. We can now factor out \((x - 2)\) from the polynomial using synthetic division or polynomial division.<br /><br />Using synthetic division to divide \(x^3 - 2x^2 - 5x + 10\) by \((x - 2)\):<br /><br />\[<br />\begin{array}{r|rrrr}<br />2 & 1 & -2 & -5 & 10 \\<br /> & & 2 & 0 & -10 \\<br />\hline<br /> & 1 & 0 & -5 & 0 \\<br />\end{array}<br />\]<br /><br />The quotient is \(x^2 - 5\), so we have:<br />\[ x^3 - 2x^2 - 5x + 10 = (x - 2)(x^2 - 5) \]<br /><br />Thus, the completely factored form of the polynomial is:<br />\[ (x - 2)(x^2 - 5) \]<br /><br />Therefore, the correct answer is:<br />\[ \boxed{(x-2)(x^{2}-5)} \]
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