given that x = 2 is a zero of f(x) = x^3 - 3x^2 - 10x + 24, apply the factor theorem and factor f(x) into linear factors
The factor theorem states that if x = c is a zero of a polynomial f(x), then (x - c) is a factor of f(x).
Since x = 2 is a zero of f(x) = x^3 - 3x^2 - 10x + 24, we can use the factor theorem to conclude that (x - 2) is a factor of f(x).
To factor f(x) into linear factors, we can perform polynomial long division.
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x - 2 | x^3 - 3x^2 - 10x + 24
- (x^3 - 2x^2)
____________________
-x^2 - 10x
+ (x^2 - 2x)
__________________
-12x + 24
+ (-12x + 24)
_________________
0
The result of the long division is x^2 - 10, which indicates that (x - 2)(x^2 - 10) is the factored form of f(x).
To apply the factor theorem, we need to substitute the given zero, x = 2, into the polynomial f(x) and check if it evaluates to zero.
Step 1: Substitute x = 2 into f(x)
f(2) = (2)^3 - 3(2)^2 - 10(2) + 24
= 8 - 12 - 20 + 24
= 0
Since f(2) evaluates to zero, we can conclude that x - 2 is a factor of f(x) by the factor theorem.
Now, let's perform polynomial division to find the remaining factor.
Step 2: Perform polynomial division using x - 2 as the divisor.
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(x - 2) | x^3 - 3x^2 - 10x + 24
- (x^3 - 2x^2)
________________________
- x^2 - 10x
- (- x^2 + 2x)
________________________
- 8x + 24
- (- 8x + 16)
________________________
8
The remainder is 8.
Step 3: Rewrite f(x) using the divisor and the quotient obtained from the polynomial division.
f(x) = (x - 2)(x^2 - x - 8)
Now, we factored f(x) into linear factors:
f(x) = (x - 2)(x^2 - x - 8)
To apply the factor theorem, we need to verify if x = 2 is a zero of the polynomial f(x) = x^3 - 3x^2 - 10x + 24.
The factor theorem states that if x = a is a zero of a polynomial f(x), then (x - a) is a factor of f(x). In other words, if plugging in a value for x gives us a result of zero, then (x - a) will be a factor of the polynomial.
Let's substitute x = 2 into f(x):
f(2) = (2)^3 - 3(2)^2 - 10(2) + 24
= 8 - 12 - 20 + 24
= 0
Since f(2) = 0, we know that x = 2 is a zero of f(x), and therefore (x - 2) is a factor of f(x).
To factor f(x) into linear factors, we can perform polynomial division using (x - 2) as the divisor.
Using polynomial division, we have:
x^2 + x - 12
______________________
x - 2 | x^3 - 3x^2 - 10x + 24
-(x^3 - 2x^2)
____________
-x^2 - 10x
+(x^2 - 2x)
___________
-8x + 24
-(-8x + 16)
___________
8
The quotient of the polynomial division is x^2 + x - 12, and the remainder is 8. This means that we can express f(x) as:
f(x) = (x - 2)(x^2 + x - 12) + 8
Therefore, the factored form of f(x) into its linear factors is:
f(x) = (x - 2)(x + 4)(x - 3)