To use synthetic division to divide the polynomial \( 4x^3 + 10x^2 - 25x - 4 \) by \( x + 4 \), we first need to rewrite the divisor \( x + 4 \) in the form \( x - c \). Here, \( c = -4 \).
Now, we will set up synthetic division using \( -4 \) and the coefficients of the polynomial \( 4, 10, -25, -4 \).
Step 1: Set up the coefficients
The coefficients of the polynomial are:
- \( 4 \) (coefficient of \( x^3 \))
- \( 10 \) (coefficient of \( x^2 \))
- \( -25 \) (coefficient of \( x \))
- \( -4 \) (constant term)
Step 2: Perform synthetic division
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Write down \( -4 \) to the left and the coefficients to the right:
\[ \begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \ & & & & \ \end{array} \]
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Bring down the leading coefficient \( 4 \):
\[ \begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \ & & & & \ & 4 & & & \ \end{array} \]
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Multiply \( -4 \) (the value we're dividing by) by \( 4 \) and write the result under the next coefficient (which is \( 10 \)):
\[ -4 \times 4 = -16 \]
Add this to \( 10 \):
\[ 10 - 16 = -6 \]
Update the table:
\[ \begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \ & & -16 & & \ & 4 & -6 & & \ \end{array} \]
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Now, multiply \( -4 \) by \( -6 \) and write it under \( -25 \):
\[ -4 \times -6 = 24 \]
Add this to \( -25 \):
\[ -25 + 24 = -1 \]
Update the table:
\[ \begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \ & & -16 & 24 & \ & 4 & -6 & -1 & \ \end{array} \]
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Finally, multiply \( -4 \) by \( -1 \) and write it under \( -4 \):
\[ -4 \times -1 = 4 \]
Add this to \( -4 \):
\[ -4 + 4 = 0 \]
Update the table:
\[ \begin{array}{r|rrrr} -4 & 4 & 10 & -25 & -4 \ & & -16 & 24 & 4 \ & 4 & -6 & -1 & 0 \ \end{array} \]
Step 3: Write the final result
The bottom row (except for the remainder) gives us the coefficients of the quotient polynomial:
The quotient polynomial is:
\[ 4x^2 - 6x - 1 \]
The remainder is \( 0 \).
Final Answer
Thus, the result of dividing \( 4x^3 + 10x^2 - 25x - 4 \) by \( x + 4 \) is:
\[ 4x^2 - 6x - 1 \] with a remainder of \( 0 \).