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
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Answer:The factors of f(x) are (x + 4) , (x - 3) , (x² + 1)The values of x = -4 and 3Step-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 ________________Step 2 : Bring down the first coefficient to the bottom row. -4 1 1 -11 1 -12 ________________ 1Step 3 : Multiply it by -4, and carry the result into the next column. -4 1 1 -11 1 -12 __-4_____________ 1Step 4 : Add down the column -4 1 1 -11 1 -12 __-4_____________ 1 -3Step 5 : Multiply it by -4, and carry the result into the next column -4 1 1 -11 1 -12 __-4__12___________ 1 -3Step 6 : Add down the column -4 1 1 -11 1 -12 __-4__12___________ 1 -3 1Step 7 : Multiply it by -4, and carry the result into the next column -4 1 1 -11 1 -12 __-4__12__-4_________ 1 -3 1Step 8 : Add down the column -4 1 1 -11 1 -12 __-4__12__-4_________ 1 -3 1 -3Step 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 -3Step 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 = 3Step 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 _____________ 1Step 3 : Multiply it by 3, and carry the result into the next column. 3 1 -3 1 -3 ____3________ 1Step 4 : Add down the column 3 1 -3 1 -3 ___3____________ 1 0Step 5 : Multiply it by 3, and carry the result into the next column 3 1 -3 1 -3 ___3__0___ 1 0Step 6 : Add down the column 3 1 -3 1 -3 ___3__0___ 1 0 1Step 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