To perform synthetic division of the polynomial \(4x^4 + 8x^3 - 15x^2 - 19x - 30\) by the linear divisor \(x + 3\), we first set \(x + 3 = 0\) which gives us \(x = -3\).
Now, we will use synthetic division with -3 as the divisor. The coefficients of the polynomial are:
\[ 4, 8, -15, -19, -30 \]
We will set up our synthetic division as follows:
-3 | 4 8 -15 -19 -30
|
|
Now, we bring down the leading coefficient (4):
-3 | 4 8 -15 -19 -30
|
|__________
4
Next, we multiply -3 by 4 and write it under the next coefficient (8):
-3 | 4 8 -15 -19 -30
| -12
|__________
4 -4
Now, we carry this process forward. We multiply -3 by -4 (the result we just got) and write that underneath the next coefficient (-15):
-3 | 4 8 -15 -19 -30
| -12 12
|__________
4 -4 -3
Next, we multiply -3 by -3 and add it to -19:
-3 | 4 8 -15 -19 -30
| -12 12 9
|__________
4 -4 -3 -10
Finally, we multiply -3 by -10 and add it to -30:
-3 | 4 8 -15 -19 -30
| -12 12 9 30
|__________
4 -4 -3 -10 0
The last row gives us the coefficients of the quotient and the remainder. The coefficients we obtained are \(4, -4, -3, -10\), and the remainder is \(0\).
Thus, the result of the division is:
\[ 4x^3 - 4x^2 - 3x - 10 \]
So, when \(4x^4 + 8x^3 - 15x^2 - 19x - 30\) is divided by \(x + 3\), the quotient is:
\[ 4x^3 - 4x^2 - 3x - 10 \]
And the remainder is \(0\).
1 answer