Question

Calculate � ( − 15 ) P(−15)​ using synthetic division and the Remainder Theorem P(x)=x 4 −4x 3 −19x 2 −46x+120

Answer 1

Using the synthetic division and remainder theorem,
`(x^4-4x^3-19x^2-46x+120)/(x+15)`
`= x^3 -19x^2+266x-4036+(60660)/(x+15)`.

Calculating polynomial function using synthetic division and remainder theorem.

We are to divide the function
`(x^4-4x^3-19x^2-46x+120)/(x+15)` using synthetic division method.

We will have to take the constant of the divisor with an opposite sign and also write the coefficients of the dividends to the right. i.e.

|| x⁴ x³ x² x¹ x⁰

-15 || 1 -4 -19 -46 120

We will write the first coefficient without changes:

|| x⁴ x³ x² x¹ x⁰

-15 || 1 -4 -19 -46 120

The next step is to multiply the entry in the left part of the table by the last entry in the result row, then add the result gotten to the next coefficient of the dividend, and write down the sum.

-15 || 1 -4 -19 -46 120

|| (-15) × (1) = -15

1 (-4) + (-15) = -19

We will repeat the same process for the remaining values, and we have:

-15 || 1 -4 -19 -46 120

|| -15 (-15) × (-19) = 285

1 -19 (-19) + 285 = 266

-15 || 1 -4 -19 -46 120

|| -15 285 (-15) × (266) = -3990

1 -19 266 (-46)+(-3990) = -4036

-15 || 1 -4 -19 -46 120

|| -15 285 -3990 (-15) × (-4036) = 60540

1 -19 266 -4036 (120)+(60540) = 60660

Now, the complete table is done and have the following resulting coefficients: 1, −19, 266, −4036, 60660. Therefore, the quotient is:

`x^3 -19x^2+266x-4036` and the remainder is 60660.

We can conclude that using the synthetic division and remainder theorem,
`(x^4-4x^3-19x^2-46x+120)/(x+15)`
`= x^3 -19x^2+266x-4036+(60660)/(x+15)`