Answer :
Final answer:
To solve the equation, we can use synthetic division and polynomial division. The given equation 45x³ + 69x² - 13x - 5 = 0 can be factored as (x + 53)(45x² - 2226x + 117822) = 0, and the zeroes are x = -53, x ≈ -20.59, and x ≈ 57.06.
Explanation:
To solve the equation 45x³ + 69x² - 13x - 5 = 0, we are given that -53 is a zero of the polynomial function f(x) = 45x³ + 69x² - 13x - 5. Since -53 is a zero, it means that (x + 53) is a factor of the polynomial. This can be verified using synthetic division or polynomial long division.
Performing synthetic division with -53 as the divisor, we get the divisor -53, the coefficients 45, 69, -13, and -5, and the calculated values 0, -2385, 126990, -6597615. Since the result of the synthetic division is 0, this confirms that -53 is indeed a zero or root of the polynomial function.
Now, we can use polynomial division to divide the given polynomial by (x + 53). The quotient will be a quadratic polynomial. Dividing 45x³ + 69x² - 13x - 5 by (x + 53), we get 45x² - 2226x + 117822 as the quotient.
Therefore, the given equation 45x³ + 69x² - 13x - 5 = 0 can be factored as (x + 53)(45x² - 2226x + 117822) = 0. To find the remaining solutions, we can set each factor equal to zero and solve for x. Setting x + 53 = 0 gives x = -53, and solving 45x² - 2226x + 117822 = 0 using the quadratic formula gives two more solutions: x ≈ -20.59 and x ≈ 57.06.
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