Find all zeros of the function f(x)=16x^3-48x^2-145x-75
Please help ASAP!!! :(
Start with factor of -75
f(x)=16x^3-48x^2-145x-75
f(1) = 16 - 48 - 145 - 75 ≠ 0
f(-1) = -16 - 48 + 145 - 75 ≠0
f(3) ≠ 0
f(-3) ≠ 0
f(5) = 0 , yeah!!! , so x-5 is a factor
using synthetic division by x-5
we have 16x^3-48x^2-145x-75 = (x-5)(16x^2 + 32x - 15)
so you have x = 5 and the two roots of 16x^2 + 32x - 15 = 0
I will leave it up to you to find those two roots, let me know what you get.
hint: they are rational.
I got -3/4 and -5/4
Good job. You found Reiny's typo.
To find the zeros of a function, set the function equal to zero and solve for x. In this case, you need to find the zeros of the function f(x) = 16x^3 - 48x^2 - 145x - 75.
Step 1: Set f(x) equal to zero:
16x^3 - 48x^2 - 145x - 75 = 0
Step 2: You can use various methods to solve for the zeros. One approach is to use synthetic division, which helps identify factors that might yield zero remainders.
Start by listing all the possible rational zeros of the function, which are the factors of the constant term (75) divided by the factors of the leading coefficient (16). In this case, the factors of 75 are ±1, ±3, ±5, ±15, ±25, and ±75, and the factors of 16 are ±1, ±2, ±4, ±8, and ±16. Therefore, the possible rational zeros are ±1, ±3/2, ±5/2, ±3, ±5, ±15/2, ±25/2, ±15, ±25, ±75/2, and ±75.
Step 3: Use synthetic division to evaluate each potential rational zero to see if it produces a remainder of zero. Starting with one of the potential rational zeros, perform synthetic division to check if it gives a remainder of zero when divided by the function f(x).
For example, let's try x = 1 as a potential rational zero:
1 | 16 -48 -145 -75
|_____ 16 -32 -113
16 -32 -113 -188
The remainder is -188, which is not zero. Therefore, x = 1 is not a zero of the function.
Repeat the synthetic division process for each potential rational zero until you find one or more zeros that produce a remainder of zero. These zeros will be the actual solutions for the function.
Continue this process until all the zeros are found. In this case, the zeros are: x = -1/2, x = -3, and x = 5/2.
Therefore, the zeros of the function f(x) = 16x^3 - 48x^2 - 145x - 75 are x = -1/2, x = -3, and x = 5/2.