Question 1170794: WRITING POLYNOMIAL FUNCTIONS Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: -2, -1, 2, 3, sqrt11.
Please show work.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Here's how to construct the polynomial function step-by-step:
**1. Understand Conjugate Pairs**
* Since the polynomial has rational coefficients, any irrational roots must come in conjugate pairs.
* Therefore, if √11 is a root, then -√11 must also be a root.
**2. Form Linear Factors**
* For each root 'r', we can create a linear factor (x - r).
* Roots: -2, -1, 2, 3, √11, -√11
* Factors: (x + 2), (x + 1), (x - 2), (x - 3), (x - √11), (x + √11)
**3. Multiply the Factors**
* To get the polynomial, we multiply these factors together.
* f(x) = (x + 2)(x + 1)(x - 2)(x - 3)(x - √11)(x + √11)
**4. Simplify by Grouping**
* Multiply the conjugate root factors first:
* (x - √11)(x + √11) = x² - (√11)² = x² - 11
* Multiply the remaining factors:
* (x + 2)(x - 2) = x² - 4
* (x + 1)(x - 3) = x² - 2x - 3
* Now, multiply these results:
* f(x) = (x² - 4)(x² - 2x - 3)(x² - 11)
**5. Expand the Polynomial**
* Multiply the first two quadratic factors:
* (x² - 4)(x² - 2x - 3) = x⁴ - 2x³ - 3x² - 4x² + 8x + 12
* = x⁴ - 2x³ - 7x² + 8x + 12
* Multiply the result by the remaining quadratic factor:
* f(x) = (x⁴ - 2x³ - 7x² + 8x + 12)(x² - 11)
* = x⁶ - 2x⁵ - 7x⁴ + 8x³ + 12x² - 11x⁴ + 22x³ + 77x² - 88x - 132
* = x⁶ - 2x⁵ - 18x⁴ + 30x³ + 89x² - 88x - 132
**Final Answer**
The polynomial function f(x) is:
f(x) = x⁶ - 2x⁵ - 18x⁴ + 30x³ + 89x² - 88x - 132
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