Answer :
Final answer:
To solve the polynomial equation 4x³ − 11x² − 54x + 45 = 0, we can determine if 5 is a zero by substituting it into the function. Then, using polynomial long division or synthetic division, we can find the quadratic equation resulting from dividing by (x - 5). Solving the resulting quadratic equation gives us the solutions for the original equation.
Explanation:
To solve the polynomial equation 4x³ – 11x² − 54x + 45 = 0, we are given that 5 is a zero of the function f(x) = 4x³ − 11x² − 54x + 45. Since 5 is a zero, it means when we substitute x = 5 into the function, the result will be 0. So, let's substitute:
- f(5) = 4(5)³ − 11(5)² − 54(5) + 45 = 500 - 275 - 270 + 45 = 0
Therefore, we have confirmed that 5 is indeed a zero of the given function. Now, we need to solve the equation 4x³ – 11x² − 54x + 45 = 0 completely. To do this, we can use polynomial long division or synthetic division to find the quadratic equation that results from dividing the original cubic equation by (x - 5). Dividing the equation, we get:
(x - 5)(4x² + 9x - 9) = 0
Now we solve the quadratic equation 4x² + 9x - 9 = 0. You can use factoring, the quadratic formula, or completing the square to solve it. By using the quadratic formula, we get:
- x = (-9 ± √(9² - 4(4)(-9))) / (2(4))
- x = (-9 ± √(81 + 144)) / 8
- x = (-9 ± √(225)) / 8
- x = (-9 ± 15) / 8
- x = (-9 + 15) / 8 or x = (-9 - 15) / 8
- x = 6 / 8 or x = -24 / 8
- x = 3/4 or x = -3
Therefore, the solutions to the equation 4x³ – 11x² − 54x + 45 = 0 are x = 5 (given) and x = 3/4 or x = -3.
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