Answer:
f(x) = (x +2)(4x -1)(x -3)
Step-by-step explanation:
The easy way
The graph shows the zeros to be x = -2, x = 1/4, and x = 3. The given function has a leading coefficient of 4, so the factorization is ...
f(x) = 4(x -(-2))(x -1/4)(x -3)
Multiplying the second term by 4, we can write the factorization as ...
f(x) = (x +2)(4x -1)(x -3)
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The hard way
Descarte's rule of signs tells you there are 0 or 2 positive real roots, and 1 negative real root. The sum of coefficients is -13 and the y-intercept (constant) is 6, so we know one of the roots is between 0 and 1.
The rational root theorem tells you the possibilities are 1/4 or 1/2. The x-intercept seems closer to x=0 (where f(0)=6) than to x=1 (where f(1)=-13), so our first guess could be x=1/4 is a root.
Synthetic division with 1/4 as a root gives the other factor as 4x^2 -4x -24, so the factorization at this point is ...
f(x) = (x -1/4)(4x^2 -4x -24) = (4x -1)(x^2 -x -6)
We can easily factor the quadratic as (x -3)(x +2), so the full factorization is ...
f(x) = (4x -1)(x -3)(x +2)
