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Chapter 2: Problem 3

Find the zeros of the polynomial function and state the multiplicity of each zero. $$P(x)=x^{2}(3 x+5)^{2}$$

Short Answer

Expert verified
The polynomial function has two zeros: \(x = 0\) with a multiplicity of 2, and \(x = -5/3\) with a multiplicity of 2.

Step by step solution

01

Identify Each Factor in the Polynomial

Here the polynomial is expressed as a product of two factors: \(x^2\) and \((3x+5)^2\). Each of these factors, when equal to zero, gives a zero of the polynomial function.
02

Find Zeros for the First Factor

For the first factor, set \(x^2 = 0\). To solve this, take the square root of both sides. This gives \(x = 0\). This zero is of multiplicity 2, because the factor \(x^2\) is associated with an exponent of 2.
03

Find Zeros for the Second Factor

For the second factor, set \((3x + 5)^2 = 0\). To solve this, take the square root of both sides to get \(3x + 5 = 0\). Solving this gives \(x = -5/3\). This zero also has a multiplicity of 2, because the factor \((3x+5)^2\) is associated with an exponent of 2.

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Most popular questions from this chapter

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