Which of the following is a factor of x^3 - 5x^2 - 2x + 24, given that x - 3 is one of the linear factors?
x - 2
x + 4
x + 2
x^2 - 8x + 22
To solve this problem, we can use the Remainder Theorem to determine which of the options is a factor of the given polynomial.
According to the Remainder Theorem, if x - a is a factor of a polynomial, then plugging in a for x will make the remainder equal to zero.
We are given that x - 3 is a factor of the polynomial x^3 - 5x^2 - 2x + 24. Plugging in x = 3 into the polynomial, we get:
(3)^3 - 5(3)^2 - 2(3) + 24
= 27 - 45 - 6 + 24
= 0
Since the remainder is zero when x = 3, it means that x - 3 is a factor of the polynomial.
Next, we can divide the polynomial x^3 - 5x^2 - 2x + 24 by x - 3 to find the quotient. The quotient will be a polynomial of degree 2 that we can factor further.
Using polynomial long division or synthetic division, we find that:
x^3 - 5x^2 - 2x + 24 รท (x - 3) = x^2 - 2x - 8
Now, we can factor x^2 - 2x - 8 to find the remaining factors. Factoring x^2 - 2x - 8, we get:
(x - 4)(x + 2)
Therefore, the factors of x^3 - 5x^2 - 2x + 24 are:
(x - 3)(x - 4)(x + 2)
Out of the given options, the only factor is x + 2. Therefore, the answer is x + 2.
To determine which of the given options is a factor of the polynomial x^3 - 5x^2 - 2x + 24, we can make use of the factor theorem. According to the factor theorem, if (x - a) is a factor of a polynomial, then plugging in the value a into the polynomial will result in zero.
Given that x - 3 is a linear factor, we can verify if it is indeed a factor by substituting 3 into the polynomial:
(3)^3 - 5(3)^2 - 2(3) + 24 = 27 - 45 - 6 + 24 = 0
Since substituting x = 3 results in zero, we can conclude that (x - 3) is a factor of the polynomial.
Now, let's check the remaining options:
1. x - 2: Substituting x = 2 into the polynomial:
(2)^3 - 5(2)^2 - 2(2) + 24 = 8 - 20 - 4 + 24 = 8
Since the result is not zero, x - 2 is not a factor.
2. x + 4: Substituting x = -4 into the polynomial:
(-4)^3 - 5(-4)^2 - 2(-4) + 24 = -64 - 80 + 8 + 24 = -112
Since the result is not zero, x + 4 is not a factor.
3. x + 2: Substituting x = -2 into the polynomial:
(-2)^3 - 5(-2)^2 - 2(-2) + 24 = -8 - 20 + 4 + 24 = 0
Since substituting x = -2 results in zero, we can conclude that (x + 2) is a factor of the polynomial.
4. x^2 - 8x + 22: This is a quadratic factor and cannot be considered as a linear factor. Therefore, it is not a factor of the polynomial.
From the given options, the factor of x^3 - 5x^2 - 2x + 24, in addition to x - 3, is (x + 2).
To find the factor of a given polynomial, you can use polynomial long division or synthetic division. In this case, since x - 3 is given as one of the linear factors, we can use synthetic division to check which of the choices is also a factor.
Let's perform synthetic division to determine if any of the choices are factors:
First, set up the synthetic division table:
3 | 1 - 5 - 2 24
Bring down the 1:
3 | 1 - 5 - 2 24
1
Multiply the divisor 3 by the number in the quotient (1):
3 | 1 - 5 - 2 24
1
------
3
Subtract the product from the second column:
3 | 1 - 5 - 2 24
1
------
3 - 15
Bring down the next coefficient -2:
3 | 1 - 5 - 2 24
1 -12
------
3 - 15 - 12
Add the numbers in the third column:
3 | 1 - 5 - 2 24
1 -12
------
3 - 15 - 12 12
Bring down the next coefficient 24:
3 | 1 - 5 - 2 24
1 -12
------
3 - 15 - 12 12
24
Since the remainder is 24, the given factor (x - 3) is not a factor of x^3 - 5x^2 - 2x + 24. Therefore, none of the choices is a factor.