To perform synthetic division of \( x^4 + 3x^2 + x - 4 \) by \( x + 3 \), we need to set up the synthetic division process and handle the missing coefficients. Since the polynomial is of degree 4, we first need the coefficients of the polynomial in descending order of the powers of \( x \). The polynomial \( x^4 + 3x^2 + x - 4 \) has the coefficients:
- \( 1 \) for \( x^4 \)
- \( 0 \) for \( x^3 \) (since there is no \( x^3 \) term)
- \( 3 \) for \( x^2 \)
- \( 1 \) for \( x^1 \)
- \( -4 \) for the constant term
So we can write the coefficients as:
\[ 1, 0, 3, 1, -4 \]
We use the root from \( x + 3 = 0 \), which is \( x = -3 \).
Now we set up the synthetic division:
-3 | 1 0 3 1 -4
| -3 9 -36 105
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1 -3 12 -35 101
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Bring down the leading coefficient (1).
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Multiply by \(-3\) (the root) and add to the next coefficient:
- \( 0 + (-3) = -3 \)
- Multiply \(-3\) by \(-3\): \( 9 \)
- \( 3 + 9 = 12 \)
- Multiply \( 12 \) by \(-3\): \( -36 \)
- \( 1 - 36 = -35 \)
- Multiply \(-35\) by \(-3\): \( 105 \)
- \( -4 + 105 = 101 \)
Thus, the result of synthetic division gives us the coefficients of the quotient polynomial, which are \( 1, -3, 12, -35 \) with a remainder of \( 101 \).
Therefore, the quotient polynomial is
\[ x^3 - 3x^2 + 12x - 35 \]
The missing terms in the quotient are:
- The coefficient for \( x^3 \) is \( 1 \).
- The coefficient for \( x^2 \) is \( -3 \).
- The coefficient for \( x^1 \) is \( 12 \).
- The constant term is \( -35 \).
So you would enter the coefficients as:
1, -3, 12, -35.