Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 27

Chapter 4, Problem 27

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91

Verified step by step guidance

Identify the given zeros of the polynomial. Since the polynomial has real coefficients and one zero is complex (4 + 3i), its complex conjugate (4 - 3i) must also be a zero. So, the zeros are -5, 4 + 3i, and 4 - 3i.

Write the factors corresponding to each zero. For zero -5, the factor is $(x + 5)$. For zeros $4 + 3i$ and $4 - 3i$, the factors are $(x - (4 + 3i))$ and $(x - (4 - 3i))$ respectively.

Multiply the complex conjugate factors to get a quadratic factor with real coefficients: \[(x - (4 + 3i))(x - (4 - 3i)) = (x - 4 - 3i)(x - 4 + 3i)\] Use the difference of squares formula to simplify this product.

Express the polynomial function as $f(x) = a(x + 5)(x^2 - 8x + 25)$, where $a$ is a real number constant to be determined.

Use the given function value $f(2) = 91$ to find $a$. Substitute $x = 2$ into the polynomial and set the expression equal to 91, then solve for $a$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Zeros and Complex Conjugates

For polynomials with real coefficients, non-real zeros always come in conjugate pairs. Given a zero like 4 + 3i, its conjugate 4 - 3i must also be a zero. This ensures the polynomial remains with real coefficients when expanded.

Constructing a Polynomial from Zeros

A polynomial can be formed by multiplying factors corresponding to its zeros. For zeros r1, r2, and r3, the polynomial is f(x) = a(x - r1)(x - r2)(x - r3), where 'a' is a leading coefficient determined by additional conditions.

Using Function Values to Determine Leading Coefficient

Given a specific function value like f(2) = 91, substitute x = 2 into the polynomial expression and solve for the leading coefficient 'a'. This step ensures the polynomial satisfies all given conditions.

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