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Answer :
To find the factors of the polynomial [tex]\(x^3 - 3x^2 + 16x - 48\)[/tex], we can follow these steps:
1. Try factoring by substitution or synthetic division: Check if there are any obvious rational roots using the Rational Root Theorem, which states that any rational root, expressed in the reduced form, is a factor of the constant term divided by a factor of the leading coefficient. For this polynomial, if we try evaluating simple numbers like [tex]\(x = 1, -1, 2, -2, 3, -3\)[/tex], we find that [tex]\(x = 3\)[/tex] is a root.
2. Factor out the root: Since [tex]\(x = 3\)[/tex] is a root, it means [tex]\(x - 3\)[/tex] is a factor of the polynomial. We can use synthetic division or polynomial division to divide the original polynomial by [tex]\(x - 3\)[/tex].
3. Polynomial division: Dividing [tex]\(x^3 - 3x^2 + 16x - 48\)[/tex] by [tex]\(x - 3\)[/tex] gives us a quotient of [tex]\(x^2 + 16\)[/tex].
4. Factor completely: After division, we find that the polynomial factors into [tex]\((x - 3)(x^2 + 16)\)[/tex]. These are the factors of the polynomial.
5. Conclusion: The factors of the polynomial [tex]\(x^3 - 3x^2 + 16x - 48\)[/tex] are [tex]\(x - 3\)[/tex] and [tex]\(x^2 + 16\)[/tex].
Thus, the polynomial can be expressed as a product of [tex]\((x - 3)\)[/tex] and [tex]\((x^2 + 16)\)[/tex]. Since [tex]\(x^2 + 16\)[/tex] is not factorable further using real numbers, these are the complete factors of this polynomial.
1. Try factoring by substitution or synthetic division: Check if there are any obvious rational roots using the Rational Root Theorem, which states that any rational root, expressed in the reduced form, is a factor of the constant term divided by a factor of the leading coefficient. For this polynomial, if we try evaluating simple numbers like [tex]\(x = 1, -1, 2, -2, 3, -3\)[/tex], we find that [tex]\(x = 3\)[/tex] is a root.
2. Factor out the root: Since [tex]\(x = 3\)[/tex] is a root, it means [tex]\(x - 3\)[/tex] is a factor of the polynomial. We can use synthetic division or polynomial division to divide the original polynomial by [tex]\(x - 3\)[/tex].
3. Polynomial division: Dividing [tex]\(x^3 - 3x^2 + 16x - 48\)[/tex] by [tex]\(x - 3\)[/tex] gives us a quotient of [tex]\(x^2 + 16\)[/tex].
4. Factor completely: After division, we find that the polynomial factors into [tex]\((x - 3)(x^2 + 16)\)[/tex]. These are the factors of the polynomial.
5. Conclusion: The factors of the polynomial [tex]\(x^3 - 3x^2 + 16x - 48\)[/tex] are [tex]\(x - 3\)[/tex] and [tex]\(x^2 + 16\)[/tex].
Thus, the polynomial can be expressed as a product of [tex]\((x - 3)\)[/tex] and [tex]\((x^2 + 16)\)[/tex]. Since [tex]\(x^2 + 16\)[/tex] is not factorable further using real numbers, these are the complete factors of this polynomial.
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