Consider the following polynomial function
F(x)=-2x^3-x^2+13x-6 to answer the following question.
If (x+3) is a factor of f(x) rewrite f(x) in completely factored form and identify the zeros f(x).
To determine if (x+3) is a factor of f(x), we can use synthetic division.
Using synthetic division, we divide f(x) by (x+3):
-3 | -2 -1 13 -6
| 6 -15 6
-------------------
-2 5 -2 0
The result of the synthetic division is -2x^2 + 5x - 2, with a remainder of 0. Since the remainder is 0, we can conclude that (x+3) is indeed a factor of f(x).
Thus, we can rewrite f(x) in completely factored form as:
f(x) = (x+3)(-2x^2 + 5x - 2)
To find the zeros of f(x), we set each factor equal to zero:
x+3 = 0,
x = -3
-2x^2 + 5x - 2 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
By factoring:
(-2x + 1)(x - 2) = 0
Setting each factor equal to zero:
-2x + 1 = 0,
x = 1/2
x - 2 = 0,
x = 2
Therefore, the zeros of f(x) are x = -3, x = 1/2, and x = 2.
To determine if (x+3) is a factor of f(x), we can use the remainder theorem. According to the theorem, if (x+3) is a factor of f(x), then f(-3) should equal 0.
Let's test this by substituting x=-3 into the polynomial function:
F(-3) = -2(-3)^3 - (-3)^2 + 13(-3) - 6
= -2(-27) - 9 + (-39) - 6
= 54 - 9 - 39 - 6
= 0
Since F(-3) equals 0, we can conclude that (x+3) is indeed a factor of f(x).
To rewrite f(x) in completely factored form, we need to rewrite f(x) as the product of its factors.
Since (x+3) is a factor of f(x), we can divide f(x) by (x+3) to determine the remaining factors.
Using long division or synthetic division, we can divide f(x) by (x+3):
-2x^2 - 7x + 2
___________________
x + 3 | -2x^3 - x^2 + 13x - 6
-(-2x^3 - 6x^2)
________________
5x^2 + 13x
-(5x^2 + 15x)
______________
-2x + 6
- (-2x -6)
______________
12
The result of the division is 12.
Now, we can express f(x) in completely factored form:
f(x) = (x+3)(-2x^2 - 7x + 2)
To further factor the quadratic expression -2x^2 - 7x + 2, we can use factoring methods such as grouping, factoring by grouping, or quadratic formula. However, in this case, it does not factor further using integers.
To find the zeros of f(x), we set each factor equal to zero and solve for x:
x + 3 = 0 => x = -3
Since (x+3) is a factor of f(x), and we have factored f(x) into (x+3)(-2x^2 - 7x + 2), the zero of f(x) is x = -3.
well, just divide. You will see that (x+3) divides it evenly, and the quotient is -2x^2+5x-2)
so, F(x) = -(x+2)(2x^2-5x+2)
= -(x+3)(x-2)(2x-1)
The zeros can now just be read off.