Math / Algebra

QuestionEvaluate the polynomial function using Synthetic Division $\begin{array}{c} f(x)=5 x^{3}+3 x^{2}-x+7 \\ \text { when } x=2 \end{array}$

Studdy Solution

STEP 1

1. We are given the polynomial function $f(x) = 5x^3 + 3x^2 - x + 7$ .
2. We need to evaluate the polynomial at $x = 2$ using synthetic division.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Interpret the result to find $f(2)$ .

STEP 3

Write down the coefficients of the polynomial: $5, 3, -1, 7$ .
Set up the synthetic division with $x = 2$ as the divisor:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & & & \\ \end{array}$

STEP 4

Bring down the leading coefficient (5):
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & & & \\ & 5 & & & \\ \end{array}$
Multiply the divisor (2) by the number just written below the line (5), and write the result (10) under the next coefficient (3):
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & & \\ & 5 & & & \\ \end{array}$
Add the numbers in the second column:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & & \\ & 5 & 13 & & \\ \end{array}$

STEP 5

Repeat the process: Multiply the divisor (2) by the new number below the line (13), and write the result (26) under the next coefficient (-1):
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & 26 & \\ & 5 & 13 & & \\ \end{array}$
Add the numbers in the third column:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & 26 & \\ & 5 & 13 & 25 & \\ \end{array}$

STEP 6

Repeat the process one more time: Multiply the divisor (2) by the new number below the line (25), and write the result (50) under the next coefficient (7):
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & 26 & 50 \\ & 5 & 13 & 25 & \\ \end{array}$
Add the numbers in the fourth column:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & 10 & 26 & 50 \\ & 5 & 13 & 25 & 57 \\ \end{array}$

STEP 7

The last number (57) is the remainder and represents the value of $f(2)$ .
Thus, $f(2) = 57$ .
The value of $f(2)$ is:
$\boxed{57}$

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QuestionEvaluate the polynomial function using Synthetic Division $\begin{array}{c} f(x)=5 x^{3}+3 x^{2}-x+7 \\ \text { when } x=2 \end{array}$

Studdy Solution

STEP 1

1. We are given the polynomial function $f(x) = 5x^3 + 3x^2 - x + 7$ .
2. We need to evaluate the polynomial at $x = 2$ using synthetic division.

STEP 2

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Interpret the result to find $f(2)$ .

STEP 3

Write down the coefficients of the polynomial: $5, 3, -1, 7$ .
Set up the synthetic division with $x = 2$ as the divisor:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & & & \\ \end{array}$

STEP 4

STEP 5

STEP 6

STEP 7

The last number (57) is the remainder and represents the value of $f(2)$ .
Thus, $f(2) = 57$ .
The value of $f(2)$ is:
$\boxed{57}$

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QuestionEvaluate the polynomial function using Synthetic Division f(x)=5x3+3x2−x+7 when x=2\begin{array}{c} f(x)=5 x^{3}+3 x^{2}-x+7 \\ \text { when } x=2 \end{array}f(x)=5x3+3x2−x+7 when x=2​

Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=5x3+3x2−x+7 f(x) = 5x^3 + 3x^2 - x + 7 f(x)=5x3+3x2−x+7.2. We need to evaluate the polynomial at x=2 x = 2 x=2 using synthetic division.

STEP 2

1. Set up the synthetic division.2. Perform the synthetic division.3. Interpret the result to find f(2) f(2) f(2).

STEP 3

Write down the coefficients of the polynomial: 5,3,−1,7 5, 3, -1, 7 5,3,−1,7.Set up the synthetic division with x=2 x = 2 x=2 as the divisor:253−17\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & & & \\ \end{array}2​5​3​−1​7​

STEP 4

STEP 5

STEP 6

STEP 7

The last number (57) is the remainder and represents the value of f(2) f(2) f(2).Thus, f(2)=57 f(2) = 57 f(2)=57. The value of f(2) f(2) f(2) is:57 \boxed{57} 57​

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QuestionEvaluate the polynomial function using Synthetic Division $\begin{array}{c} f(x)=5 x^{3}+3 x^{2}-x+7 \\ \text { when } x=2 \end{array}$

1. We are given the polynomial function $f(x) = 5x^3 + 3x^2 - x + 7$ .
2. We need to evaluate the polynomial at $x = 2$ using synthetic division.

1. Set up the synthetic division.
2. Perform the synthetic division.
3. Interpret the result to find $f(2)$ .

Write down the coefficients of the polynomial: $5, 3, -1, 7$ .
Set up the synthetic division with $x = 2$ as the divisor:
$\begin{array}{c|cccc} 2 & 5 & 3 & -1 & 7 \\ & & & & \\ \end{array}$

The last number (57) is the remainder and represents the value of $f(2)$ .
Thus, $f(2) = 57$ .
The value of $f(2)$ is:
$\boxed{57}$