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O POLYNOMIAL AND RATIONAL FUNCTIONSFinding zeros and their multiplicities given a polynomial functio.

O POLYNOMIAL AND RATIONAL FUNCTIONSFinding zeros and their multiplicities given a-example-1
User Didxga
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2 Answers

7 votes
7 votes

Final answer:

The solution of quadratic equations generally involves finding zeros and understanding their multiplicities in the context of second-order polynomials. Graphing polynomials helps visualize the impact of coefficients on the curve's shape. Equation graphing tools are useful in exploring these concepts interactively.

Step-by-step explanation:

The solution of quadratic equations involves finding the zeros of a quadratic function, which is a type of second-order polynomial. These functions can be expressed in the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. To find zeros, one can factor the quadratic, complete the square, or use the quadratic formula, which is x = (-b ± √(b² - 4ac)) / (2a). The multiplicity of a zero refers to the number of times that zero appears as a root of the polynomial.

Additionally, understanding the graphing of polynomials is essential to visualize the curve's shape, which changes as the coefficients are adjusted. This aspect is particularly important when examining the effects of different terms in the polynomial on the overall graph.

The Equation Grapher tools, such as those provided by PhET Explorations, can help visualize how individual terms contribute to the overall shape of the polynomial curve. This interactive learning allows a deeper understanding of how changes in the polynomial coefficients affect its graph.

User JSager
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25 votes
25 votes

From the function, we can conclude:


\begin{gathered} (x+6)^3 \\ x=-6_{\text{ }}Multiplicity_{\text{ }}3 \\ ---------- \\ (x-9)^2 \\ x=9_{\text{ }}Multiplicity_{\text{ }}2 \\ ----------- \\ x-6 \\ x=6_{\text{ }}Multiplicity_{\text{ }}1 \\ ----------- \\ (x+4)^3 \\ x=-4_{\text{ }}Multiplicity_{\text{ }}3 \end{gathered}

Answer:

Zero(s) of multiplicity one: 6

Zero(s) of multiplicity two: 9

Zero(s) of multiplicity three: -6, -4

User Bandula Dharmadasa
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