Concept explainers
To write : the polynomial as the product of linear factors and list all the zeros of function
Answer to Problem 71E
The function as product of linear factors:
The zeros of the function are:
Explanation of Solution
Given information:
Concept Involved:
Linear Factorization Theorem:If
Complex Zeros Occur in Conjugate Pairs: Let f be a polynomial function that has real coefficients. If
Synthetic Division (for a Cubic Polynomial):To divide
In case when we have a polynomial with a missing term, insert placeholders with zero coefficients for missing powers of the variable. Vertical pattern: Add terms in columns Diagonal pattern: Multiply results by k. This algorithm for synthetic division works only for divisors of the form x - k. Remember that |
The Division Algorithm: If
Graph:
Interpretation:
From the graph of the function we can pick possible zeros of the function as
Calculation:
Use synthetic division to find the other zeros of the function
If
To find other zeros of the polynomial
We can solve this equation using completing the square method by subtracting 5 on both sides of the equation
Simplify on both sides of the equation
In order to make the left side expression as perfect square trinomial, we need to add square of half of coefficient of x on both sides
Simplify on right side of the equation
Write the left side as a perfect square
Take square root on both sides
Simplifying square root on both sides of the equation
Replace
Add 2 on both sides of the equation
Simplify in left side of the equation
List the zeros of the functions given:
If
So
Write the function
Conclusion:
The zeros of the given function
The function written as product of linear factors:
Chapter 2 Solutions
EBK PRECALCULUS W/LIMITS
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning