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QuestionDivide the polynomial x3+9x2+32x+48x^{3}+9 x^{2}+32 x+48 by x+4x+4 using synthetic division.

Studdy Solution

STEP 1

Assumptions1. We are given a cubic polynomial x3+9x+32x+48x^{3}+9 x^{}+32 x+48 and we are asked to divide it by a linear polynomial x+4x+4. . We are asked to use synthetic division to solve this problem.

STEP 2

First, we need to set up the synthetic division. Write down the coefficients of the cubic polynomial and the root of the linear polynomial. The root of x+4x+4 is −4-4.
−4193248\begin{array}{c|cccc} -4 &1 &9 &32 &48 \\ \end{array}

STEP 3

Now, bring down the first coefficient (1) to the bottom row.
−1932481\begin{array}{c|cccc} - &1 &9 &32 &48 \\ &1 & & & \\ \end{array}

STEP 4

Multiply the value we just brought down (1) by the root of the divisor (-4), and write the result under the second coefficient of the dividend (9).
−41932481−4\begin{array}{c|cccc} -4 &1 &9 &32 &48 \\ &1 & -4 & & \\ \end{array}

STEP 5

Add the values in the second column (9 and -4), and write the result in the bottom row.
−419324815\begin{array}{c|cccc} -4 &1 &9 &32 &48 \\ &1 &5 & & \\ \end{array}

STEP 6

Repeat the process for the remaining columns. Multiply the value we just calculated (5) by the root of the divisor (-4), and write the result under the third coefficient of the dividend (32).
−419324815−20\begin{array}{c|cccc} -4 &1 &9 &32 &48 \\ &1 &5 & -20 & \\ \end{array}

STEP 7

Add the values in the third column (32 and -20), and write the result in the bottom row.
−41932481512\begin{array}{c|cccc} -4 &1 &9 &32 &48 \\ &1 &5 &12 & \\ \end{array}

STEP 8

Again, multiply the value we just calculated (12) by the root of the divisor (-4), and write the result under the fourth coefficient of the dividend (48).
−4132481512−48\begin{array}{c|cccc} -4 &1 & &32 &48 \\ &1 &5 &12 & -48 \\ \end{array}

STEP 9

Finally, add the values in the fourth column (48 and -48), and write the result in the bottom row.
−493248512\begin{array}{c|cccc} -4 & &9 &32 &48 \\ & &5 &12 & \\ \end{array}

STEP 10

The bottom row of coefficients represents the quotient polynomial. The degree of the quotient polynomial is one less than the degree of the dividend, so the quotient is a quadratic polynomial.Quotient=x2+5x+12Quotient = x^{2} +5x +12The remainder is0, which means x3+9x2+32x+48x^{3}+9 x^{2}+32 x+48 is exactly divisible by x+4x+4.

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