Studdy AI Logo
Math

Question Use synthetic division to find 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-60. Simplify your answers.

Studdy Solution

STEP 1

Assumptions
1. The given function is f(x)=x3−13x2+52x−60f(x) = x^3 - 13x^2 + 52x - 60.
2. We need to find the values of f(2)f(2), f(−5)f(-5), and f(6)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)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}

STEP 3

Perform the synthetic division operations.
21−1352−602⋅1=22⋅(−11)=−222⋅30=60 \begin{array}{r|rrrr} 2 & 1 & -13 & 52 & -60 \\ \hline & & 2\cdot1=2 & 2\cdot(-11)=-22 & 2\cdot30=60 \\ \end{array}
21−13+252−22−60+60−11300 \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).
f(2)=0f(2) = 0

STEP 5

To find 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}

STEP 6

Perform the synthetic division operations.
−51−1352−60−5⋅1=−5−5⋅(−18)=90−5⋅142=−710 \begin{array}{r|rrrr} -5 & 1 & -13 & 52 & -60 \\ \hline & & -5\cdot1=-5 & -5\cdot(-18)=90 & -5\cdot142=-710 \\ \end{array}
−51−13−552+90−60−710−18142−770 \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).
f(−5)=−770f(-5) = -770

STEP 8

To find 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}

STEP 9

Perform the synthetic division operations.
61−1352−606⋅1=66⋅(−7)=−426⋅10=60 \begin{array}{r|rrrr} 6 & 1 & -13 & 52 & -60 \\ \hline & & 6\cdot1=6 & 6\cdot(-7)=-42 & 6\cdot10=60 \\ \end{array}
61−13+652−42−60+60−7100 \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).
f(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 - 60 and evaluate f(2)f(2), f(−5)f(-5), and 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 = 2, x=−5x = -5, and x=6x = 6 to verify the values obtained through synthetic division.

STEP 13

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

Was this helpful?

Studdy solves anything!

banner

Start learning now

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

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord