Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure, let's tackle this synthetic-division problem step-by-step!
Given the polynomial [tex]\( 2x^4 + x^3 - 2x^2 - 8x + 5 \)[/tex] and dividing it by [tex]\( x - 2 \)[/tex]:
1. Set up the synthetic division: Write down the coefficients of the polynomial:
[tex]\[ \begin{array}{ccccc} 2 & 1 & -2 & -8 & 5 \\ \end{array} \][/tex]
2. Divide by [tex]\( x - 2 \)[/tex]: The divisor here is 2.
3. Carry down the leading coefficient (which is 2) to the bottom row:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2 & & & & & 2 \\ \end{array} \][/tex]
4. Multiply the 2 (bottom row) by the divisor (2) and write the result underneath the next coefficient (1):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & - & 4 & & & 2 \\ \end{array} \][/tex]
Here, [tex]\( 2 \times 2 = 4 \)[/tex].
5. Add the numbers in the second column: [tex]\( 1 + 4 = 5 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & & & 2 \\ & 2 & 5 & & & 2 \\ \end{array} \][/tex]
6. Multiply the recent bottom row value by the divisor (2) and place the result in the next column:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & & 2 \\ & 2 & 5 & 8 & & 2 \\ \end{array} \][/tex]
Here, [tex]\( 5 \times 2 = 10 \)[/tex].
7. Add the numbers in the third column: [tex]\( -2 + 10 = 8 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & 2 \\ & 2 & 5 & 8 & 8 & 2 \\ \end{array} \][/tex]
8. Next, [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & & 2 \\ & 2 & 5 & 8 & 8 & 21 & 2 \\ \end{array} \][/tex]
9. Add the numbers in the fourth column: [tex]\( -8 + 16 = 8 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & & 2 \\ & 2 & 5 & 8 & 16 & 2 \\ \end{array} \][/tex]
10. Finally, multiply [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & 21 & 2 \\ \end{array} \][/tex]
11. Add the numbers in the fifth column: [tex]\( 5 + 16 = 21 \)[/tex].
The final coefficients after synthetic division are:
[tex]\[ 2, 5, 8, 8, 21 \][/tex]
So, the numbers obtained from the synthetic division are [tex]\(2, 5, 8, 8, 21\)[/tex].
Thus, the number that belongs in the place of the question mark is [tex]\(21\)[/tex].
Given the polynomial [tex]\( 2x^4 + x^3 - 2x^2 - 8x + 5 \)[/tex] and dividing it by [tex]\( x - 2 \)[/tex]:
1. Set up the synthetic division: Write down the coefficients of the polynomial:
[tex]\[ \begin{array}{ccccc} 2 & 1 & -2 & -8 & 5 \\ \end{array} \][/tex]
2. Divide by [tex]\( x - 2 \)[/tex]: The divisor here is 2.
3. Carry down the leading coefficient (which is 2) to the bottom row:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2 & & & & & 2 \\ \end{array} \][/tex]
4. Multiply the 2 (bottom row) by the divisor (2) and write the result underneath the next coefficient (1):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & - & 4 & & & 2 \\ \end{array} \][/tex]
Here, [tex]\( 2 \times 2 = 4 \)[/tex].
5. Add the numbers in the second column: [tex]\( 1 + 4 = 5 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & & & 2 \\ & 2 & 5 & & & 2 \\ \end{array} \][/tex]
6. Multiply the recent bottom row value by the divisor (2) and place the result in the next column:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & & 2 \\ & 2 & 5 & 8 & & 2 \\ \end{array} \][/tex]
Here, [tex]\( 5 \times 2 = 10 \)[/tex].
7. Add the numbers in the third column: [tex]\( -2 + 10 = 8 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & 2 \\ & 2 & 5 & 8 & 8 & 2 \\ \end{array} \][/tex]
8. Next, [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & & 2 \\ & 2 & 5 & 8 & 8 & 21 & 2 \\ \end{array} \][/tex]
9. Add the numbers in the fourth column: [tex]\( -8 + 16 = 8 \)[/tex]:
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & & 2 \\ & 2 & 5 & 8 & 16 & 2 \\ \end{array} \][/tex]
10. Finally, multiply [tex]\( 8 \times 2 = 16 \)[/tex] (place in the next column):
[tex]\[ \begin{array}{ccccc} & 2 & 1 & -2 & -8 & 5 \\ 2: & & 4 & 10 & 16 & 21 & 2 \\ \end{array} \][/tex]
11. Add the numbers in the fifth column: [tex]\( 5 + 16 = 21 \)[/tex].
The final coefficients after synthetic division are:
[tex]\[ 2, 5, 8, 8, 21 \][/tex]
So, the numbers obtained from the synthetic division are [tex]\(2, 5, 8, 8, 21\)[/tex].
Thus, the number that belongs in the place of the question mark is [tex]\(21\)[/tex].
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
