Respuesta :
Answer:
The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)
The values of x = -4 and 3
Step-by-step explanation:
* At first we calculate f(-4)
- If it will be = 0
∴ x + 4 is a factor of f(x)
* In the synthetic calculation we use the coefficient of the dividend
with the factors of the function
∵ [tex]f(x)=x^{4}+x^{3}-11x^{2}+x-12[/tex]
∵ x = -4
Step 1 : Write down the coefficients of the f(x) , put x = -4 at the left
-4 1 1 -11 1 -12
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Step 2 : Bring down the first coefficient to the bottom row.
-4 1 1 -11 1 -12
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1
Step 3 : Multiply it by -4, and carry the result into the next column.
-4 1 1 -11 1 -12
__-4_____________
1
Step 4 : Add down the column
-4 1 1 -11 1 -12
__-4_____________
1 -3
Step 5 : Multiply it by -4, and carry the result into the next column
-4 1 1 -11 1 -12
__-4__12___________
1 -3
Step 6 : Add down the column
-4 1 1 -11 1 -12
__-4__12___________
1 -3 1
Step 7 : Multiply it by -4, and carry the result into the next column
-4 1 1 -11 1 -12
__-4__12__-4_________
1 -3 1
Step 8 : Add down the column
-4 1 1 -11 1 -12
__-4__12__-4_________
1 -3 1 -3
Step 9 : Multiply it by -4 , and carry the result into the next column
-4 1 1 -11 1 -12
__-4__12__-4__12_______
1 -3 1 -3
Step 10 : Add down the column
-4 1 1 -11 1 -12
__-4__12__-4__12_______
1 -3 1 -3 0
∴ The quotient is (x³ - 3x² + x - 3) and the remainder is 0
* At second we calculate f(3)
- If it will be = 0
∴ x - 3 is a factor of f(x)
* We will do the same steps with x = 3
∵ f(x) = x³ - 3x² + x - 3
∵ x = 3
Step 1 : Write down the coefficients of the f(x) , put x = 3 at the left
3 1 -3 1 -3
____________
Step 2 : Bring down the first coefficient to the bottom row.
3 1 -3 1 -3
_____________
1
Step 3 : Multiply it by 3, and carry the result into the next column.
3 1 -3 1 -3
____3________
1
Step 4 : Add down the column
3 1 -3 1 -3
___3____________
1 0
Step 5 : Multiply it by 3, and carry the result into the next column
3 1 -3 1 -3
___3__0___
1 0
Step 6 : Add down the column
3 1 -3 1 -3
___3__0___
1 0 1
Step 7 : Multiply it by 3, and carry the result into the next column
3 1 -3 1 -3
___3__0__3_
1 0 1
Step 8 : Add down the column
3 1 -3 1 -3
___3__0__3_
1 0 1 0
∴ The quotient is (x² + 1) and the remainder is 0
* The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)
∴ The values of x = -4 and 3
