To decompose the polynomial \(x^3 - 2x^2 - 5x + 6\) into its linear factors, given that \(x - 1\) is a factor, we can use polynomial long division or synthetic division.
Step 1: Synthetic Division
Since \(x - 1\) is a factor, we can use synthetic division with \(1\):
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Write the coefficients of the polynomial: \(1, -2, -5, 6\).
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Set up the synthetic division:
1 | 1 -2 -5 6
| 1 -1 -6
-----------------
1 -1 -6 0
The bottom row shows the coefficients of the quotient polynomial, which is \(1x^2 - 1x - 6\).
Step 2: Factor the Quotient
Now, we need to factor the quadratic \(x^2 - x - 6\):
To factor \(x^2 - x - 6\), we look for two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\).
Thus:
\[ x^2 - x - 6 = (x - 3)(x + 2) \]
Step 3: Combine the Factors
The complete factorization of the original polynomial is:
\[ x^3 - 2x^2 - 5x + 6 = (x - 1)(x - 3)(x + 2) \]
Final Answer
The linear factors of the polynomial \(x^3 - 2x^2 - 5x + 6\) are:
\[ (x - 1)(x - 3)(x + 2) \]
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