# Factor the polynomial. (a) `f(x)= 4x^3 + 4x^2 - 23x - 30`I'm having trouble understanding how to factor polynomials.

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jgeertz | Certified Educator

If a polynomial function has integer coefficients, then every rational zero will be of the form p/q where p is a factor of the constant and q is a factor of tee leading coefficient. In this problem p=30 and q=4.

To find the roots of the polynomial , consider all combinations of +-p/q.

+-1, +-1/2, +-1/4, +-2, +-3, +-3/2, +-3/4, +-5, +-5/2, +-5/4, +-6, +-10, +-15,+-15/2, +-15/4, +-30.

The combination -2 works once it is substituted into the equation.

4(-2)^3 + 4(-2)^2 - 23(-2) -30

Synthetic division of (4x^3 + 4x^2 -23x -30)/ (x+2)

yields 4x^2-4x-15

The remaining roots are x=-3/2, 5/2**Now we can rewrite the polynomial as a group of factors.**

**(x+2)(x+3/2)(x-5/2) The roots of the polynomial 4x^3 + 4x^2 -23x -30 are x= -2, -3/2, 5/2 **

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