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Answer :
To factor the polynomial expression [tex]\( P(x) = 6x^3 - 19x^2 + 11x + 6 \)[/tex] completely, let's follow these steps:
1. Look for Common Factors: Check if there is a common factor for all terms first. In this case, there isn't a common factor aside from 1.
2. Use the Rational Root Theorem: This theorem can help us identify possible rational roots of the polynomial by considering factors of the constant term (6) and the leading coefficient (6).
3. Test Possible Roots: Evaluate possible rational roots using synthetic division or direct substitution to identify any roots. Possible rational roots to test could include [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
4. Factor Using Found Roots: Once a root is found, use synthetic division or polynomial division to factor the polynomial by the corresponding linear factor.
5. Repeat the Process: Continue factoring the quotient obtained until it is completely factored into irreducible factors (those that cannot be factored further over the integers).
For the given polynomial, when these steps are applied correctly, you will find that the polynomial can be factored as:
[tex]\[ P(x) = (2x - 3)(3x + 2)(x - 1) \][/tex]
This means that the polynomial is completely factored into the product of three linear terms. Each of these factors corresponds to a root of the original cubic polynomial. This step-by-step factorization process ensures that the polynomial is fully simplified into its simplest components.
1. Look for Common Factors: Check if there is a common factor for all terms first. In this case, there isn't a common factor aside from 1.
2. Use the Rational Root Theorem: This theorem can help us identify possible rational roots of the polynomial by considering factors of the constant term (6) and the leading coefficient (6).
3. Test Possible Roots: Evaluate possible rational roots using synthetic division or direct substitution to identify any roots. Possible rational roots to test could include [tex]\( \pm 1, \pm 2, \pm 3, \pm 6 \)[/tex].
4. Factor Using Found Roots: Once a root is found, use synthetic division or polynomial division to factor the polynomial by the corresponding linear factor.
5. Repeat the Process: Continue factoring the quotient obtained until it is completely factored into irreducible factors (those that cannot be factored further over the integers).
For the given polynomial, when these steps are applied correctly, you will find that the polynomial can be factored as:
[tex]\[ P(x) = (2x - 3)(3x + 2)(x - 1) \][/tex]
This means that the polynomial is completely factored into the product of three linear terms. Each of these factors corresponds to a root of the original cubic polynomial. This step-by-step factorization process ensures that the polynomial is fully simplified into its simplest components.
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