To decompose the polynomial \((x^2 - 7x + 6)(x^2 + 3x - 18)\) into linear factors, we will first factor each quadratic polynomial individually.
Step 1: Factor \(x^2 - 7x + 6\)
We need to find two numbers that multiply to \(6\) (the constant term) and add up to \(-7\) (the coefficient of \(x\)). The numbers that fit this criteria are \(-6\) and \(-1\).
Thus, we can factor: \[ x^2 - 7x + 6 = (x - 6)(x - 1) \]
Step 2: Factor \(x^2 + 3x - 18\)
Next, we need to find two numbers that multiply to \(-18\) (the constant term) and add up to \(3\) (the coefficient of \(x\)). The numbers that fit this criteria are \(6\) and \(-3\).
Thus, we can factor: \[ x^2 + 3x - 18 = (x + 6)(x - 3) \]
Step 3: Combine the factors
Now, we can combine both factorizations together: \[ (x^2 - 7x + 6)(x^2 + 3x - 18) = (x - 6)(x - 1)(x + 6)(x - 3) \]
So, the polynomial \((x^2 - 7x + 6)(x^2 + 3x - 18)\) decomposes into linear factors: \[ \boxed{(x - 6)(x - 1)(x + 6)(x - 3)} \]
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