To factor the expression g(x) = 3x^2 + 10x − 8 and find the zeros of g(x), you can use the following steps:
- First, you need to find two numbers that multiply to give the product of the coefficient of x^2 and the constant term, which is 3 × (−8) = −24, and add to give the coefficient of x, which is 10. (In brief basically find out a pair that you can multiply to get 3 times -8 (-24) and can add to get 10. One possible pair of numbers is 12 and −2, since 12 × (−2) = −24 and 12 + (−2) = 10.
- Next, you need to rewrite the expression g(x) by splitting the middle term 10x into 12x and −2x, using the pair of numbers you found. This gives:
g(x) = 3x^2 + 10x - 8
g(x) = 3x^2 + 12x - 2x - 8
- Then, you need to factor the expression g(x) by grouping the first two terms and the last two terms separately, and taking out the common factors. This gives:
g(x) = 3x(x + 4) - 2(x + 4)
- Finally, you need to factor the expression g(x) by noticing that (x + 4) is a common factor for both terms, and using the distributive property in reverse. This gives:
g(x) = (3x - 2)(x + 4)
This is the factored form of g(x).
To find the zeros of g(x), you need to set g(x) equal to zero and solve for x. This gives:
g(x) = 0
(3x - 2)(x + 4) = 0
Using the zero product property, we get:
3x - 2 = 0 \text{ or } x + 4 = 0
Solving for x, we get:
x = {2}/{3} or x = -4
These are the zeros of g(x).
I hope this answer is helpful to you. If you have any other questions, feel free to ask.
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