Find all zeros of $x^3-5 x^2+x-5$, graphed below. Enter the zeros separated by commas.
Answer (2)
Factor the polynomial by grouping: ( x 3 − 5 x 2 ) + ( x − 5 ) = x 2 ( x − 5 ) + 1 ( x − 5 ) = ( x − 5 ) ( x 2 + 1 ) .
Set the factored polynomial to zero: ( x − 5 ) ( x 2 + 1 ) = 0 .
Solve for x : x − 5 = 0 ⇒ x = 5 and x 2 + 1 = 0 ⇒ x = ± i .
The zeros are 5 , i , − i , so the final answer is 5 , i , − i .
Explanation
Understanding the Problem We are given the polynomial x 3 − 5 x 2 + x − 5 and asked to find its zeros. This means we need to find the values of x for which the polynomial equals zero.
Factoring by Grouping To find the zeros, we can try to factor the polynomial. We can use factoring by grouping. Group the first two terms and the last two terms: ( x 3 − 5 x 2 ) + ( x − 5 ) .
Factoring out Common Factors Factor out x 2 from the first group and 1 from the second group: x 2 ( x − 5 ) + 1 ( x − 5 ) .
Factoring out (x-5) Now, we can factor out the common factor ( x − 5 ) from both terms: ( x − 5 ) ( x 2 + 1 ) .
Setting the Polynomial to Zero Set the factored polynomial equal to zero: ( x − 5 ) ( x 2 + 1 ) = 0 .
Solving for x Now, we solve for x in each factor. First, x − 5 = 0 , which gives us x = 5 . Second, x 2 + 1 = 0 , which gives us x 2 = − 1 . Taking the square root of both sides, we get x = ± i .
Final Answer Therefore, the zeros of the polynomial are 5 , i , − i .
Examples
Polynomials and their zeros are fundamental in many areas of mathematics and engineering. For example, in control systems, the roots of the characteristic equation determine the stability of a system. Finding the zeros of a polynomial allows engineers to design stable and predictable systems, such as cruise control in a car or the autopilot system in an airplane. Factoring polynomials helps in simplifying complex expressions and solving equations that arise in various scientific and engineering applications.
The zeros of the polynomial x 3 − 5 x 2 + x − 5 are 5 , i , − i . We arrived at this by factoring the polynomial and solving each factor for zero. Thus, the complete solution is 5 , i , − i .
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