Which of the following is a zero of x^3−5x^2−2x+24 , given that x−3 is one of the linear factors?(1 point)
1. -4
2. 2
3. -2
4. -8
First, we know that x-3 is a factor of x^3 - 5x^2 - 2x + 24 because x = 3 makes the expression equal to zero.
To find the other zero, we can use synthetic division or long division to divide x^3 - 5x^2 - 2x + 24 by x - 3.
Using synthetic division:
3 | 1 -5 -2 24
3 -6 -24
-----------------
1 -2 -8 0
The result is 1 - 2x - 8, and since the remainder is 0, this tells us that x - 3 is a factor of x^3 - 5x^2 - 2x + 24, and x = 3 is one of the zeros.
The remaining polynomial, 1 - 2x - 8, can be factored as (x + 4)(x - 2).
Therefore, the zeros of the polynomial are x = 3, x = -4, and x = 2.
So, the correct answer is:
1. -4
To find a zero of a polynomial, we need to determine the value of x that makes the polynomial equal to zero.
First, we are given that x - 3 is one of the linear factors of the polynomial x^3 - 5x^2 - 2x + 24. This means that when x is equal to 3, the polynomial becomes zero.
So, substitute x = 3 into the polynomial:
(3)^3 - 5(3)^2 - 2(3) + 24 = 0
27 - 45 - 6 + 24 = 0
0 = 0
Since the polynomial equals 0 when x = 3, we have found one zero.
To find the remaining zeros, we can divide the given polynomial by (x - 3). This can be done using polynomial long division or synthetic division.
Using synthetic division:
3 | 1 - 5 - 2 24
| 3 -6 -12
________________
1 - 2 - 8 12
After performing the division, we obtain the quotient x^2 - 2x - 8.
Factoring the quotient:
(x^2 - 2x - 8) = (x + 2)(x - 4)
From this factorization, we can see that two additional zeros are x = -2 and x = 4.
Therefore, the zeros of the given polynomial x^3 - 5x^2 - 2x + 24 are:
1. x = 3 (given)
2. x = -2 (from the factorization)
3. x = 4 (from the factorization)
None of the given options (-4, 2, -2, -8) matches the zeros we found.
To find the zero/roots of a polynomial, we need to set the polynomial equal to zero. In this case, the given polynomial is:
x^3 - 5x^2 - 2x + 24
We are also given that x - 3 is a linear factor of the polynomial. This means that when we divide the polynomial by x - 3, the result will be a quadratic polynomial. Let's perform the long division:
x^2 - 2x - 8
____________________
x - 3 | x^3 - 5x^2 - 2x + 24
- x^3 + 3x^2
_______________
- 8x^2 - 2x
+ 8x^2 - 24x
_______________
-14x + 24
+14x - 42
______________
66
The result of the long division is:
(x^3 - 5x^2 - 2x + 24) / (x - 3) = (x^2 - 2x - 8) + (66 / (x - 3))
From this result, we can see that (x^2 - 2x - 8) is the quadratic polynomial obtained by dividing the original polynomial by x - 3.
Now, to find the zeros of the polynomial, we need to solve the quadratic equation:
x^2 - 2x - 8 = 0
Using factoring or the quadratic formula, we find that the solutions to this equation are x = -2 and x = 4.
Therefore, the zero/roots of the polynomial x^3 - 5x^2 - 2x + 24, given that x - 3 is one of the linear factors, are:
-2 and 4
Therefore, the answer is:
Option: 2. 2