Find $h$ so that $x+5$ is a factor of $x^4+6 x^3+9 x^2+h x+20$.
Answer (2)
Apply the factor theorem: substitute x = − 5 into the polynomial and set it equal to 0.
Simplify the equation: ( − 5 ) 4 + 6 ( − 5 ) 3 + 9 ( − 5 ) 2 + h ( − 5 ) + 20 = 0 becomes 120 − 5 h = 0 .
Solve for h : 5 h = 120 , so h = 24 .
The value of h is 24 .
Explanation
Understanding the Problem We are given that x + 5 is a factor of the polynomial x 4 + 6 x 3 + 9 x 2 + h x + 20 . We need to find the value of h .
Applying the Factor Theorem Since x + 5 is a factor of the polynomial x 4 + 6 x 3 + 9 x 2 + h x + 20 , then by the factor theorem, if we plug in x = − 5 into the polynomial, it should equal 0.
Substituting the Value So, we substitute x = − 5 into the polynomial: ( − 5 ) 4 + 6 ( − 5 ) 3 + 9 ( − 5 ) 2 + h ( − 5 ) + 20 = 0
Simplifying the Equation Now, let's simplify the equation: 625 + 6 ( − 125 ) + 9 ( 25 ) − 5 h + 20 = 0 625 − 750 + 225 − 5 h + 20 = 0 120 − 5 h = 0
Solving for h Now, we solve for h : − 5 h = − 120 h = − 5 − 120 h = 24
Final Answer Therefore, the value of h is 24.
Examples
Polynomial factorization is used in cryptography to design secure communication protocols. By representing data as polynomials, cryptographic algorithms can leverage the mathematical properties of polynomials to encrypt and decrypt information. The process of finding factors of a polynomial is crucial in breaking or strengthening these cryptographic systems, making it a fundamental concept in cybersecurity.
To find the value of h so that x + 5 is a factor of the polynomial x 4 + 6 x 3 + 9 x 2 + h x + 20 , we use the Factor Theorem. Substituting x = − 5 into the polynomial and simplifying leads us to conclude that h = 24 . Thus, the answer is h = 24 .
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