Find all the zeros of the function and write the polynomial as a product of linear factors.
g(x)=3x^3-4x^2+8x+8
On my graphing calculator it says that it is -2/3. But when I do it by hand using synthetic division, I don't get a zero. I'm going crazy here... I've tried -2/3 and 2/3 none of them work... But when I look on my graphing calculator it says that it is -2/3. Help?
if x= -2/3 then one of the factors is 3x+2
I did algebraic long division and got x^2 - 2x + 4 with no remainder.
so g(x)=3x^3-4x^2+8x+8
= (3x+2)(x^2-2x+4)
so x= -2/3 or x = 1 ± √-3 wich are complex numbers
you will not be able to express it as a product of only linear factors, but rather as a
product of a linear factor times a quadratic factor
BTW, I did get a zero when doing synthetic division.
(-2/3) │ 3 -4 8 8
********** -2 4 -8
******** 3 -6 12 0
To find the zeros of the function g(x) = 3x^3 - 4x^2 + 8x + 8, you can use various methods, such as factoring, the Rational Root theorem, or synthetic division. Let's go through each step to find the zeros and write the polynomial as a product of linear factors.
1. Factoring: Unfortunately, the given polynomial is not easily factorable by finding common factors or using simple factoring techniques.
2. Rational Root theorem: The Rational Root theorem states that if a polynomial has a rational root (a zero) in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient, then p is a factor of the constant term (in this case, 8) and q is a factor of the leading coefficient (in this case, 3).
To apply the Rational Root theorem, you would need to consider all possible rational roots by creating fractions from the factors of 8. The factors of 8 are ±1, ±2, ±4, and ±8, and the factors of 3 are ±1 and ±3.
However, in this case, none of the factors of 8 divided by the factors of 3 (such as -8/3, 8/3, -4/3, 4/3, etc.) are zeros of the given polynomial.
3. Synthetic division: Synthetic division is another method to find the zeros of a polynomial. Let's use this method to verify the zero suggested by your graphing calculator, -2/3.
To perform synthetic division:
-2/3 | 3 -4 8 8
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Following the synthetic division process, we start by writing the coefficients of the polynomial in a row, excluding the powers of x.
1st row: 3 -4 8 8
Then, bring down the first coefficient (3) directly below the horizontal line.
2nd row: 3
-2/3
Next, multiply the -2/3 (our supposed zero) by the coefficient 3 and write the result below the next coefficient (-4). Add the numbers in the same column.
2nd row: 3
-2/3 4
Repeat this process for all the remaining coefficients, multiplying the result by -2/3 and adding it to the next coefficient.
2nd row: 3
-2/3 4
_______________
3 -2 12
The number at the lower right corner, 12, represents the remainder. Hence, our divisor -2/3 is not a zero of the given polynomial since the remainder is nonzero.
Therefore, the value -2/3 is not a zero of g(x) = 3x^3 - 4x^2 + 8x + 8.
If your graphing calculator shows -2/3 as a zero of the function, it is likely due to an approximation error or an incorrect setting.