Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
4. Polynomial Functions
Dividing Polynomials
Problem 55
Textbook Question
Use synthetic division to determine whether the given number k is a zero of the polyno-mial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x^4 + 2x^3 -x+3; k=2/5
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1
Write down the coefficients of the polynomial \( f(x) = 5x^4 + 2x^3 + 0x^2 - x + 3 \) as [5, 2, 0, -1, 3].
Set up the synthetic division by writing \( k = \frac{2}{5} \) to the left and the coefficients to the right.
Bring down the leading coefficient (5) to the bottom row.
Multiply \( \frac{2}{5} \) by the number just written on the bottom row (5) and write the result under the next coefficient (2).
Add the numbers in the second column (2 and the result from the previous step) and write the sum in the bottom row. Repeat the multiply and add process for the remaining coefficients.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Synthetic Division
Synthetic division is a simplified method for dividing a polynomial by a linear binomial of the form (x - c). It allows for quicker calculations compared to long division, focusing on the coefficients of the polynomial. This technique is particularly useful for evaluating polynomials at specific values and determining if those values are roots of the polynomial.
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Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is ƒ(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where 'n' is a non-negative integer and 'a' are constants. Understanding the structure of polynomial functions is essential for analyzing their behavior, including finding zeros and evaluating function values.
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Zero of a Polynomial
A zero of a polynomial is a value of x for which the polynomial evaluates to zero, meaning ƒ(x) = 0. Finding zeros is crucial for understanding the roots of the polynomial, which can indicate where the graph intersects the x-axis. If a given number k is not a zero, calculating ƒ(k) provides insight into the polynomial's value at that point, which is important for further analysis.
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Related Practice