To decompose the polynomial \( x^3 - 2x^2 - 5x + 6 \) into its linear factors, given that \( x - 1 \) is a factor, we can perform polynomial long division.
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Divide the polynomial \( x^3 - 2x^2 - 5x + 6 \) by \( x - 1 \):
Step 1: Divide the leading term of the polynomial by the leading term of the divisor: \[ \frac{x^3}{x} = x^2 \]
Step 2: Multiply \( x^2 \) by \( x - 1 \): \[ x^2(x - 1) = x^3 - x^2 \]
Step 3: Subtract this from the original polynomial: \[ (x^3 - 2x^2 - 5x + 6) - (x^3 - x^2) = -2x^2 + x^2 - 5x + 6 = -x^2 - 5x + 6 \]
Step 4: Repeat the process with the new polynomial \(-x^2 - 5x + 6\): Divide the leading term: \[ \frac{-x^2}{x} = -x \]
Multiply: \[ -x(x - 1) = -x^2 + x \]
Subtract: \[ (-x^2 - 5x + 6) - (-x^2 + x) = -5x - x + 6 = -6x + 6 \]
Step 5: Divide \(-6x + 6\): \[ \frac{-6x}{x} = -6 \]
Multiply: \[ -6(x - 1) = -6x + 6 \]
Subtract: \[ (-6x + 6) - (-6x + 6) = 0 \]
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The result of the division is: \[ x^3 - 2x^2 - 5x + 6 = (x - 1)(x^2 - x - 6) \]
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Now, factor \( x^2 - x - 6 \): We need two numbers that multiply to \(-6\) and add to \(-1\). The pair that works is \( -3 \) and \( 2 \): \[ x^2 - x - 6 = (x - 3)(x + 2) \]
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Combine the factors: Therefore, the complete factorization of the polynomial is: \[ x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2) \]
Thus, the polynomial \( x^3 - 2x^2 - 5x + 6 \) decomposes into its linear factors as: \[ (x - 1)(x - 3)(x + 2) \]
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