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Solve the equation [tex]45x^3 + 69x^2 - 13x - 5 = 0[/tex] given that [tex]-\frac{5}{3}[/tex] is a zero of [tex]f(x) = 45x^3 + 69x^2 - 13x - 5[/tex].

Answer :

Main Answer:

To solve the equation 45x³+69x²-13x-5=0 given that -(5/3) is a zero of f(x)=45x³+69x²-13x-5, you can use synthetic division and the factor theorem to find the remaining quadratic factor.

Explanation:

To solve the equation 45x³+69x²-13x-5=0 when -(5/3) is a zero of the function f(x)=45x³+69x²-13x-5, we first perform synthetic division. Synthetic division is a method used to divide a polynomial by a linear factor. In this case, we divide f(x) by (3x+5), which is the linear factor corresponding to the given zero.

Using synthetic division, we get:

[tex]-5/3 | 45 69 -13 -5 ----------------- 45 0 0 0[/tex]

The result is a quotient of 45x², which represents the remaining quadratic factor. Now, we have the equation 45x²=0. To find the solutions for this quadratic equation, we set it equal to zero and solve for x:

45x² = 0

Dividing both sides by 45:

x² = 0

Taking the square root of both sides:

x = 0

So, the remaining solutions for the original cubic equation are x = -(5/3) and x = 0.

In conclusion, the solutions to the equation 45x³+69x²-13x-5=0, given that -(5/3) is a zero, are x = -(5/3) and x = 0.

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