Use synthetic division and the Remainder Theorem to find the indicated function value.

Ask by Carter Mckenzie. in the United States

Mar 20,2025

Question

Use synthetic division and the Remainder Theorem to find the indicated function value. $$ f(x)=5 x^{3}-11 x^{2}+3 x-3 ; f(1) $$ $$ f(1)=\square $$

UpStudy AI · Accepted Answer

1. Write down the coefficients of $$ f(x) = 5x^3 - 11x^2 + 3x - 3 $$, which are:  
   $$
   5,\quad -11,\quad 3,\quad -3
   $$

2. Set up the synthetic division with $$ x = 1 $$:
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     &    &      &     &     \
   \end{array}
   $$

3. Bring down the first coefficient:
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  &      &     &     \
   \end{array}
   $$

4. Multiply $$ 5 $$ by $$ 1 $$ and write the result under $$-11$$:
   
   $$
   5 \times 1 = 5
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & 5    &     &     \
   \end{array}
   $$

5. Add $$-11$$ and $$5$$:  
   
   $$
   -11 + 5 = -6
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & -6   &     &     \
   \end{array}
   $$

6. Multiply $$-6$$ by $$ 1 $$:
   
   $$
   -6 \times 1 = -6
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & -6   & -6  &     \
   \end{array}
   $$

7. Add $$ 3 $$ and $$-6$$:
   
   $$
   3 + (-6) = -3
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & -6   & -3  &     \
   \end{array}
   $$

8. Multiply $$-3$$ by $$ 1 $$:
   
   $$
   -3 \times 1 = -3
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & -6   & -3  & -3  \
   \end{array}
   $$

9. Finally, add $$-3$$ and $$-3$$:
   
   $$
   -3 + (-3) = -6
   $$
   
   $$
   \begin{array}{r|cccc}
   1 & 5  & -11  & 3   & -3  \ \hline
     & 5  & -6   & -3  & -6  \
   \end{array}
   $$

The final value, which is the remainder, is $$-6$$. According to the Remainder Theorem, $$ f(1) $$ is equal to this remainder. Thus:

$$
f(1) = -6
$$
$$ f(1) = -6 $$

Use synthetic division and the Remainder Theorem to find the indicated function value.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions