If a polynomial p(x) is given by p(x) = x^3 - 6x^2 + 11x - 6, what can be inferr

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Question: If a polynomial p(x) is given by p(x) = x^3 - 6x^2 + 11x - 6, what can be inferred about its roots?

Options:

Correct Answer: It has three distinct real roots.

Solution:

By applying the Rational Root Theorem and synthetic division, we can find that p(x) has three distinct real roots.

If a polynomial p(x) is given by p(x) = x^3 - 6x^2 + 11x - 6, what can be inferred about its roots?

If a polynomial p(x) is given by p(x) = x^3 - 6x^2 + 11x - 6, what can be inferred about its roots?

Step 1: Identify the polynomial p(x) = x^3 - 6x^2 + 11x - 6.
Step 2: Use the Rational Root Theorem to find possible rational roots. The possible rational roots are the factors of the constant term (-6) divided by the factors of the leading coefficient (1).
Step 3: List the factors of -6: ±1, ±2, ±3, ±6.
Step 4: Test each possible rational root by substituting them into p(x) to see if they equal zero.
Step 5: Use synthetic division to divide p(x) by any root found in Step 4 to simplify the polynomial.
Step 6: Repeat the process with the resulting polynomial until all roots are found.
Step 7: Conclude that p(x) has three distinct real roots based on the results from synthetic division.

Polynomial Roots – Understanding the nature and number of roots of a polynomial function.
Rational Root Theorem – A method to identify possible rational roots of a polynomial.
Synthetic Division – A technique used to divide polynomials and find roots.