Math  /  Algebra

QuestionUse synthetic division to find the quotient and remainder when 2x4+9x3+x28x+72 x^{4}+9 x^{3}+x^{2}-8 x+7 is divided by x+4x+4 by completing the parts below. (a) Complete this synthetic division table. \begin{tabular}{llllll} \hline 4)) & 2 & 9 & 1 & -8 & 7 \end{tabular} \square \square \square \square \square \square \square \square \square (b) Write your answer in the following form: Quotient + Remainder x+4+\frac{\text { Remainder }}{x+4}. 2x4+9x3+x28x+7x+4=+x+4\frac{2 x^{4}+9 x^{3}+x^{2}-8 x+7}{x+4}=\square+\frac{\square}{x+4} Explimition Check © 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use I Privacy Center I Accessibility

Studdy Solution

STEP 1

What is this asking? We're going to divide a polynomial, 2x4+9x3+x28x+72x^4 + 9x^3 + x^2 - 8x + 7, by x+4x + 4 using synthetic division, and then write the result as a quotient plus a remainder. Watch out! Remember that synthetic division uses the *root* of the divisor, so we'll use 4-4 not +4+4!

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Write the quotient and remainder.

STEP 3

The polynomial we're dividing is 2x4+9x3+x28x+72x^4 + 9x^3 + x^2 - 8x + 7.
So, our coefficients are 2\mathbf{2}, 9\mathbf{9}, 1\mathbf{1}, 8\mathbf{-8}, and 7\mathbf{7}.

STEP 4

Since we're dividing by x+4x + 4, the root we'll use for synthetic division is 4\mathbf{-4} (because x+4=0x + 4 = 0 gives us x=4x = -4).

STEP 5

Let's set up our synthetic division table with the root and coefficients:
429187\begin{array}{c} -4 \\ 2 \\ 9 \\ 1 \\ -8 \\ 7 \\ \end{array}

STEP 6

Bring down the 2\mathbf{2}:
4291872\begin{array}{cccccc} -4 & 2 & 9 & 1 & -8 & 7 \\ & 2 \end{array}

STEP 7

* Multiply 4-4 by 2\mathbf{2} to get 8\mathbf{-8}, and add it to 9\mathbf{9}: 9+(8)=19 + (-8) = \mathbf{1}.
429187\begin{array}{c|ccccc} -4 & 2 & 9 & 1 & -8 & 7 \\ \end{array}* Multiply 4-4 by 1\mathbf{1} to get 4\mathbf{-4}, and add it to 1\mathbf{1}: 1+(4)=31 + (-4) = \mathbf{-3}.
429187213\begin{array}{cccccc} -4 & 2 & 9 & 1 & -8 & 7 \\ & 2 & 1 & -3 \end{array}* Multiply 4-4 by 3\mathbf{-3} to get 12\mathbf{12}, and add it to 8\mathbf{-8}: 8+12=4-8 + 12 = \mathbf{4}.
4291872134\begin{array}{cccccc} -4 & 2 & 9 & 1 & -8 & 7 \\ & 2 & 1 & -3 & 4 \end{array}* Multiply 4-4 by 4\mathbf{4} to get 16\mathbf{-16}, and add it to 7\mathbf{7}: 7+(16)=97 + (-16) = \mathbf{-9}.
42918721349\begin{array}{cccccc} -4 & 2 & 9 & 1 & -8 & 7 \\ 2 & 1 & -3 & 4 & -9 \end{array}

STEP 8

The numbers 2\mathbf{2}, 1\mathbf{1}, 3\mathbf{-3}, and 4\mathbf{4} are the coefficients of the quotient, and 9\mathbf{-9} is the remainder.

STEP 9

Since we started with an x4x^4 term and divided by an xx term, the quotient starts with an x3x^3 term.
So, the quotient is 2x3+x23x+42x^3 + x^2 - 3x + 4, and the remainder is 9-9.

STEP 10

2x4+9x3+x28x+7x+4=2x3+x23x+4+9x+4\frac{2x^4 + 9x^3 + x^2 - 8x + 7}{x + 4} = 2x^3 + x^2 - 3x + 4 + \frac{-9}{x + 4}

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