Question

Synthetic calculation of f(x) = x^4+x^3-11x^2+x-12 Value of x= -4 and x=3. how to solve the above question

Answer 1

The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)

The values of x = -4 and 3

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

∵
`f(x)=x^(4)+x^(3)-11x^(2)+x-12`

∵ x = -4

Step 1 : Write down the coefficients of the f(x) , put x = -4 at the left

-4 1 1 -11 1 -12

________________

Step 2 : Bring down the first coefficient to the bottom row.

-4 1 1 -11 1 -12

________________

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