Solved on Mar 18, 2024

Use synthetic division to find $f(2), f(-5), f(6)$ for $f(x)=x^{3}-13 x^{2}+52 x-60$ . Simplify your answers.

STEP 1

Assumptions
1. The given function is $f(x) = x^3 - 13x^2 + 52x - 60$ .
2. We need to find the values of $f(2)$ , $f(-5)$ , and $f(6)$ using synthetic division.
3. After finding the values, we will check the results using a graphing calculator.

STEP 2

To find $f(2)$ , we set up synthetic division with 2 as the root we are testing.
$\begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2 & -22 & 60 \\ \end{array}$

STEP 3

Perform the synthetic division operations.
$\begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2\cdot1=2 & 2\cdot(-11)=-22 & 2\cdot30=60 \\ \end{array}$
$\begin{array}{r|rrrr} 2 & 1 & -13+2 & 52-22 & -60+60 \\ \hline & & -11 & 30 & 0 \\ \end{array}$

STEP 4

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(2)$ .
$f(2) = 0$

STEP 5

To find $f(-5)$ , we set up synthetic division with -5 as the root we are testing.
$\begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5 & 90 & -460 \\ \end{array}$

STEP 6

Perform the synthetic division operations.
$\begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5\cdot1=-5 & -5\cdot(-18)=90 & -5\cdot142=-710 \\ \end{array}$
$\begin{array}{r|rrrr} -5 & 1 & -13-5 & 52+90 & -60-710 \\ \hline & & -18 & 142 & -770 \\ \end{array}$

STEP 7

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(-5)$ .
$f(-5) = -770$

STEP 8

To find $f(6)$ , we set up synthetic division with 6 as the root we are testing.
$\begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6 & -42 & 60 \\ \end{array}$

STEP 9

Perform the synthetic division operations.
$\begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6\cdot1=6 & 6\cdot(-7)=-42 & 6\cdot10=60 \\ \end{array}$
$\begin{array}{r|rrrr} 6 & 1 & -13+6 & 52-42 & -60+60 \\ \hline & & -7 & 10 & 0 \\ \end{array}$

STEP 10

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(6)$ .
$f(6) = 0$

STEP 11

To check our work, we use a graphing calculator to plot the function $f(x) = x^3 - 13x^2 + 52x - 60$ and evaluate $f(2)$ , $f(-5)$ , and $f(6)$ .

STEP 12

After plotting the function on a graphing calculator, we find the points on the graph corresponding to $x = 2$ , $x = -5$ , and $x = 6$ to verify the values obtained through synthetic division.

STEP 13

The graphing calculator should confirm that $f(2) = 0$ , $f(-5) = -770$ , and $f(6) = 0$ .
The function values are: $f(2) = 0$ $f(-5) = -770$ $f(6) = 0$

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Use synthetic division to find $f(2), f(-5), f(6)$ for $f(x)=x^{3}-13 x^{2}+52 x-60$ . Simplify your answers.

STEP 1

Assumptions
1. The given function is $f(x) = x^3 - 13x^2 + 52x - 60$ .
2. We need to find the values of $f(2)$ , $f(-5)$ , and $f(6)$ using synthetic division.
3. After finding the values, we will check the results using a graphing calculator.

STEP 2

To find $f(2)$ , we set up synthetic division with 2 as the root we are testing.
$\begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2 & -22 & 60 \\ \end{array}$

STEP 3

Perform the synthetic division operations.
$\begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2\cdot1=2 & 2\cdot(-11)=-22 & 2\cdot30=60 \\ \end{array}$
$\begin{array}{r|rrrr} 2 & 1 & -13+2 & 52-22 & -60+60 \\ \hline & & -11 & 30 & 0 \\ \end{array}$

STEP 4

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(2)$ .
$f(2) = 0$

STEP 5

To find $f(-5)$ , we set up synthetic division with -5 as the root we are testing.
$\begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5 & 90 & -460 \\ \end{array}$

STEP 6

Perform the synthetic division operations.
$\begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5\cdot1=-5 & -5\cdot(-18)=90 & -5\cdot142=-710 \\ \end{array}$
$\begin{array}{r|rrrr} -5 & 1 & -13-5 & 52+90 & -60-710 \\ \hline & & -18 & 142 & -770 \\ \end{array}$

STEP 7

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(-5)$ .
$f(-5) = -770$

STEP 8

To find $f(6)$ , we set up synthetic division with 6 as the root we are testing.
$\begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6 & -42 & 60 \\ \end{array}$

STEP 9

Perform the synthetic division operations.
$\begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6\cdot1=6 & 6\cdot(-7)=-42 & 6\cdot10=60 \\ \end{array}$
$\begin{array}{r|rrrr} 6 & 1 & -13+6 & 52-42 & -60+60 \\ \hline & & -7 & 10 & 0 \\ \end{array}$

STEP 10

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(6)$ .
$f(6) = 0$

STEP 11

To check our work, we use a graphing calculator to plot the function $f(x) = x^3 - 13x^2 + 52x - 60$ and evaluate $f(2)$ , $f(-5)$ , and $f(6)$ .

STEP 12

After plotting the function on a graphing calculator, we find the points on the graph corresponding to $x = 2$ , $x = -5$ , and $x = 6$ to verify the values obtained through synthetic division.

STEP 13

The graphing calculator should confirm that $f(2) = 0$ , $f(-5) = -770$ , and $f(6) = 0$ .
The function values are: $f(2) = 0$ $f(-5) = -770$ $f(6) = 0$

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Use synthetic division to find f(2),f(−5),f(6)f(2), f(-5), f(6)f(2),f(−5),f(6) for f(x)=x3−13x2+52x−60f(x)=x^{3}-13 x^{2}+52 x-60f(x)=x3−13x2+52x−60. Simplify your answers.

STEP 1

STEP 2

To find f(2)f(2)f(2), we set up synthetic division with 2 as the root we are testing.21−1352−602−2260 \begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2 & -22 & 60 \\ \end{array} 2​1​−132​52−22​−6060​​

STEP 3

STEP 4

Read off the remainder from the bottom row of the synthetic division. This is the value of f(2)f(2)f(2).f(2)=0f(2) = 0f(2)=0

STEP 5

To find f(−5)f(-5)f(−5), we set up synthetic division with -5 as the root we are testing.−51−1352−60−590−460 \begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5 & 90 & -460 \\ \end{array} −5​1​−13−5​5290​−60−460​​

STEP 6

STEP 7

Read off the remainder from the bottom row of the synthetic division. This is the value of f(−5)f(-5)f(−5).f(−5)=−770f(-5) = -770f(−5)=−770

STEP 8

To find f(6)f(6)f(6), we set up synthetic division with 6 as the root we are testing.61−1352−606−4260 \begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6 & -42 & 60 \\ \end{array} 6​1​−136​52−42​−6060​​

STEP 9

STEP 10

Read off the remainder from the bottom row of the synthetic division. This is the value of f(6)f(6)f(6).f(6)=0f(6) = 0f(6)=0

STEP 11

To check our work, we use a graphing calculator to plot the function f(x)=x3−13x2+52x−60f(x) = x^3 - 13x^2 + 52x - 60f(x)=x3−13x2+52x−60 and evaluate f(2)f(2)f(2), f(−5)f(-5)f(−5), and f(6)f(6)f(6).

STEP 12

After plotting the function on a graphing calculator, we find the points on the graph corresponding to x=2x = 2x=2, x=−5x = -5x=−5, and x=6x = 6x=6 to verify the values obtained through synthetic division.

STEP 13

The graphing calculator should confirm that f(2)=0f(2) = 0f(2)=0, f(−5)=−770f(-5) = -770f(−5)=−770, and f(6)=0f(6) = 0f(6)=0.The function values are: f(2)=0f(2) = 0f(2)=0 f(−5)=−770f(-5) = -770f(−5)=−770 f(6)=0f(6) = 0f(6)=0

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Use synthetic division to find $f(2), f(-5), f(6)$ for $f(x)=x^{3}-13 x^{2}+52 x-60$ . Simplify your answers.

To find $f(2)$ , we set up synthetic division with 2 as the root we are testing.
$\begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2 & -22 & 60 \\ \end{array}$

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(2)$ .
$f(2) = 0$

To find $f(-5)$ , we set up synthetic division with -5 as the root we are testing.
$\begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5 & 90 & -460 \\ \end{array}$

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(-5)$ .
$f(-5) = -770$

To find $f(6)$ , we set up synthetic division with 6 as the root we are testing.
$\begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6 & -42 & 60 \\ \end{array}$

Read off the remainder from the bottom row of the synthetic division. This is the value of $f(6)$ .
$f(6) = 0$

To check our work, we use a graphing calculator to plot the function $f(x) = x^3 - 13x^2 + 52x - 60$ and evaluate $f(2)$ , $f(-5)$ , and $f(6)$ .

After plotting the function on a graphing calculator, we find the points on the graph corresponding to $x = 2$ , $x = -5$ , and $x = 6$ to verify the values obtained through synthetic division.

The graphing calculator should confirm that $f(2) = 0$ , $f(-5) = -770$ , and $f(6) = 0$ .
The function values are: $f(2) = 0$ $f(-5) = -770$ $f(6) = 0$