Use the Factor Theorem to factor g(x)=x3−3x2−10x+24
into a product of linear factors, given that −3
is a zero of g(x)
.(1 point)
g(x)=
First, we need to find the factor of the equation g(x) = x^3 - 3x^2 - 10x + 24 by synthetic division.
-3 | 1 -3 -10 24
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1 -6 8 0
The remainder is 0, which means -3 is a zero of g(x). Therefore, (x+3) is a factor of g(x).
To find the other factor, we divide the polynomial obtained from synthetic division (1x^2 - 6x + 8) by using the factored form (x+3).
x-3
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x+3 | 1 -6 8
-(1 -3 9)
___________
-3 17
Therefore, the factorization of g(x) = x^3 - 3x^2 - 10x + 24 is (x+3)(x^2 - 3x + 8).
To use the Factor Theorem to factorize g(x), we need to find the linear factor that corresponds to the zero -3.
The Factor Theorem states that if a polynomial function f(x) has a zero of c, then (x-c) is a factor of f(x).
Since -3 is a zero of g(x), we know that (x+3) is a factor of g(x).
Now, to factorize g(x) completely, we can use long division or synthetic division to divide g(x) by (x+3):
x^2 - 6x + 8
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x+3 | x^3 - 3x^2 - 10x + 24
- (x^3 + 3x^2)
------------------
-6x^2 - 10x
+ (6x^2 + 18x)
-----------------
8x + 24
- (8x + 24)
--------------
0
The result of the division is x^2 - 6x + 8.
Therefore, we can factorize g(x) as:
g(x) = (x+3)(x^2 - 6x + 8).
To use the Factor theorem to factor a polynomial, you need to find the values (zeros) of the polynomial. Given that -3 is a zero of g(x), it means that if you substitute -3 into the polynomial, the result will be zero.
Let's substitute -3 into g(x):
g(-3) = (-3)^3 - 3(-3)^2 - 10(-3) + 24
= -27 - 3(9) + 30 + 24
= -27 - 27 + 30 + 24
= 0
Since the result is zero, it confirms that -3 is indeed a zero of g(x).
Now that we know -3 is a zero of g(x), we can use this information to factorize the polynomial. The factor theorem states that if x - a is a factor of the polynomial, then (x-a) is a factor of the polynomial.
So, we can factorize g(x) by dividing it by (x - (-3)) or (x + 3):
g(x) = (x^3 - 3x^2 - 10x + 24) / (x + 3)
Using polynomial long division or synthetic division, you can divide (x^3 - 3x^2 - 10x + 24) by (x + 3) to get the remaining factors. Once divided, you should obtain:
g(x) = (x + 3)(x^2 - 6x + 8)
Thus, g(x) can be factored as (x + 3)(x^2 - 6x + 8).