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Sagot :
We need to use synthetic division to divide the given polynomials:
[tex]\[ w + 6 \longdiv{-4w^4 - 26w^3 - 12w^2 - 3w - 36} \][/tex]
Firstly, confirm that the divisor is in the form [tex]\(w + c\)[/tex]. Here it is [tex]\(w + 6\)[/tex]. Notice that the corresponding root for this divisor is [tex]\(w = -6\)[/tex].
Next, write down the coefficients of the dividend polynomial:
[tex]\[ -4, -26, -12, -3, -36 \][/tex]
Here’s a step-by-step procedure for synthetic division:
1. Write down the root of the divisor, which is -6.
2. Set up the synthetic division using the coefficients of the dividend and the root of the divisor.
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ \end{array} \][/tex]
3. Bring down the leading coefficient, -4, directly below the line:
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & & & \\ \end{array} \][/tex]
4. Multiply -4 (the number just brought down) by the root (-6), and write the result below the next coefficient in the dividend.
[tex]\[ -4 \times -6 = 24 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & & \\ \end{array} \][/tex]
5. Add this result (24) to the next coefficient (-26):
[tex]\[ -26 + 24 = -2 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & \\ \end{array} \][/tex]
6. Repeat this process for each of the remaining coefficients:
[tex]\[ -2 \times -6 = 12 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 \\ \end{array} \][/tex]
[tex]\[ -12 + 12 = 0 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 \\ \end{array} \][/tex]
[tex]\[ 0 \times -6 = 0 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 \\ \end{array} \][/tex]
[tex]\[ -3 + 0 = -3 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 \\ \end{array} \][/tex]
[tex]\[ -3 \times -6 = 18 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 & 18\\ \end{array} \][/tex]
[tex]\[ -36 + 18 = -18 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 & 18 & -18\\ \end{array} \][/tex]
The coefficients for the quotient polynomial are the numbers we get in the row after the line excluding the last number which is our remainder.
So, the quotient is:
[tex]\[ -4w^3 - 2w^2 + 0w - 3 \][/tex]
And the remainder is:
[tex]\[ -18 \][/tex]
Thus, the quotient and remainder of the division are:
[tex]\[ \boxed{(-4w^3 - 2w^2 + 0w - 3)} \][/tex]
[tex]\[ \boxed{-18} \][/tex]
[tex]\[ w + 6 \longdiv{-4w^4 - 26w^3 - 12w^2 - 3w - 36} \][/tex]
Firstly, confirm that the divisor is in the form [tex]\(w + c\)[/tex]. Here it is [tex]\(w + 6\)[/tex]. Notice that the corresponding root for this divisor is [tex]\(w = -6\)[/tex].
Next, write down the coefficients of the dividend polynomial:
[tex]\[ -4, -26, -12, -3, -36 \][/tex]
Here’s a step-by-step procedure for synthetic division:
1. Write down the root of the divisor, which is -6.
2. Set up the synthetic division using the coefficients of the dividend and the root of the divisor.
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ \end{array} \][/tex]
3. Bring down the leading coefficient, -4, directly below the line:
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & & & \\ \end{array} \][/tex]
4. Multiply -4 (the number just brought down) by the root (-6), and write the result below the next coefficient in the dividend.
[tex]\[ -4 \times -6 = 24 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & & \\ \end{array} \][/tex]
5. Add this result (24) to the next coefficient (-26):
[tex]\[ -26 + 24 = -2 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & \\ \end{array} \][/tex]
6. Repeat this process for each of the remaining coefficients:
[tex]\[ -2 \times -6 = 12 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 \\ \end{array} \][/tex]
[tex]\[ -12 + 12 = 0 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 \\ \end{array} \][/tex]
[tex]\[ 0 \times -6 = 0 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 \\ \end{array} \][/tex]
[tex]\[ -3 + 0 = -3 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 \\ \end{array} \][/tex]
[tex]\[ -3 \times -6 = 18 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 & 18\\ \end{array} \][/tex]
[tex]\[ -36 + 18 = -18 \][/tex]
[tex]\[ \begin{array}{r|rrrrr} -6 & -4 & -26 & -12 & -3 & -36 \\ & & -4 & 24 & -2 & 12 & 0 & -3 & 18 & -18\\ \end{array} \][/tex]
The coefficients for the quotient polynomial are the numbers we get in the row after the line excluding the last number which is our remainder.
So, the quotient is:
[tex]\[ -4w^3 - 2w^2 + 0w - 3 \][/tex]
And the remainder is:
[tex]\[ -18 \][/tex]
Thus, the quotient and remainder of the division are:
[tex]\[ \boxed{(-4w^3 - 2w^2 + 0w - 3)} \][/tex]
[tex]\[ \boxed{-18} \][/tex]
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