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Answer :
The polynomial f(x) = (x – 5)⁷ (x + 1)⁵ has two zeros: x = 5 with multiplicity 7 and x = -1 with multiplicity 5.
To find the zeros of a factored polynomial and determine their multiplicities, we look at the factors and their exponents. The exponents indicate the multiplicity of each zero. The given polynomial is f(x) = (x – 5)⁷ (x + 1)⁵. This means we have two zeros: x = 5 and x = -1.
The exponent of the factor (x – 5) is 7, which tells us that x = 5 is a zero with a multiplicity of 7. Similarly, the exponent of the factor (x + 1) is 5, indicating that x = -1 is a zero with a multiplicity of 5. Therefore, the correct answer is: x = -1 with multiplicity 5, and x = 5 with multiplicity 7.
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The zeros and their multiplicities for the given factored polynomial f(x) = (x - 5)^" (x + 1) are: x = -1 with multiplicity 1, and x = 5 with multiplicity 7.
A zero of a polynomial is a value of x that makes the polynomial equal to zero.
In this case, the factored polynomial f(x) = (x - 5)^" (x + 1) is already in factored form, where each factor (x - 5) and (x + 1) represents a zero of the polynomial.
The exponent indicates the multiplicity of each zero.
From the given expression, we can see that the zero x = -1 has a multiplicity of 1, which means it appears once as a root of the polynomial.
On the other hand, the zero x = 5 has a multiplicity of 7, indicating that it appears 7 times as a root of the polynomial.
The concept of multiplicity refers to how many times a particular zero occurs as a root.
In this case, x = -1 appears once, while x = 5 appears 7 times. This information helps us understand the behavior of the polynomial near these zeros.
Zeros with higher multiplicities tend to have a stronger influence on the shape of the graph of the polynomial near those points.
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