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Determine whether the given binomial is a factor of the polynomial \( p(x) \). If it is, find the remaining factors of \( p(x) \). 2. \( p(x)=x^{3}-2 x^{2}+7 x-14 ; x-2 \)

Ask by Ford Best. in the United States
Jan 24,2025

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Answer

\( x - 2 \) is a factor of \( p(x) \), and the remaining factor is \( x^{2} + 7 \). So, \( p(x) = (x - 2)(x^{2} + 7) \).

Solution

To determine whether \( x - 2 \) is a factor of the polynomial \( p(x) = x^{3} - 2x^{2} + 7x - 14 \), and to find the remaining factors if it is, follow these steps: ### Step 1: Check if \( x - 2 \) is a Factor Use the **Factor Theorem**, which states that \( x - a \) is a factor of \( p(x) \) if and only if \( p(a) = 0 \). Evaluate \( p(2) \): \[ p(2) = (2)^3 - 2(2)^2 + 7(2) - 14 = 8 - 8 + 14 - 14 = 0 \] Since \( p(2) = 0 \), \( x - 2 \) is indeed a factor of \( p(x) \). ### Step 2: Divide \( p(x) \) by \( x - 2 \) to Find the Remaining Factor Perform polynomial division (synthetic division is efficient here): **Synthetic Division Setup:** \[ \begin{array}{c|cccc} 2 & 1 & -2 & 7 & -14 \\ \hline & & 2 & 0 & 14 \\ \hline & 1 & 0 & 7 & 0 \\ \end{array} \] **Explanation:** 1. Bring down the leading coefficient: **1**. 2. Multiply by 2: \( 1 \times 2 = 2 \), add to the next coefficient: \( -2 + 2 = 0 \). 3. Multiply by 2: \( 0 \times 2 = 0 \), add to the next coefficient: \( 7 + 0 = 7 \). 4. Multiply by 2: \( 7 \times 2 = 14 \), add to the last term: \( -14 + 14 = 0 \). The result of the division is: \[ p(x) = (x - 2)(x^2 + 7) \] ### Step 3: Factor the Quadratic (if possible) The quadratic \( x^2 + 7 \) does not factor further over the real numbers. However, if factoring over the complex numbers: \[ x^2 + 7 = (x + i\sqrt{7})(x - i\sqrt{7}) \] ### **Final Answer:** Yes, \(x - 2\) is a factor of \(p(x)\). The remaining factor is the quadratic \(x^{2} + 7\). Thus, \[ p(x) = (x - 2)(x^{2} + 7) \]

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Bonus Knowledge

To determine if \( x - 2 \) is a factor of the polynomial \( p(x) = x^{3} - 2x^{2} + 7x - 14 \), you can use the Factor Theorem, which asserts that if \( p(c) = 0 \) for some value \( c \), then \( x - c \) is a factor of \( p(x) \). Substituting \( x = 2 \) into \( p(x) \): \[ p(2) = (2)^{3} - 2(2)^{2} + 7(2) - 14 \] \[ = 8 - 8 + 14 - 14 = 0 \] Since \( p(2) = 0 \), \( x - 2 \) is indeed a factor. To find the remaining factors of \( p(x) \), you can perform polynomial long division of \( p(x) \) by \( x - 2 \). Performing the long division, we find: \[ p(x) \div (x - 2) = x^{2} + 0x + 7 = x^{2} + 7 \] Thus, we have \[ p(x) = (x - 2)(x^{2} + 7). \] The remaining factor \( x^{2} + 7 \) can’t be factored further using real numbers, but it has complex roots. Therefore, the complete factorization over the complex numbers would be: \[ p(x) = (x - 2)(x - i\sqrt{7})(x + i\sqrt{7}). \] Now, if you're feeling adventurous with polynomial factors and want to dive deeper, remember that factoring can be visualized like puzzle solving! Each polynomial can hide its secrets behind cleverly arranged terms, just waiting for a keen eye (and perhaps some division) to unravel the mystery of its composition. When working through polynomial long division, it's essential to keep your terms organized. A common mistake is to misalign degrees or skip steps, leading to confusion. Think of it as balancing a scale: every term must have its place to keep the equation's integrity intact!

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