Use synthetic division to find the function values.
find , and .
(Type an exact answer, using radicals as needed.)

Ask by Todd Johnson. in the United States

Mar 12,2025

Question

Use synthetic division to find the function values. $$ f(x)=x^{4}-16 ; $$ find $$ f(3), f(-3) $$, and $$ f(1-\sqrt{6}) $$. $$ f(3)=\square $$ (Type an exact answer, using radicals as needed.)

UpStudy AI · Accepted Answer

**Synthetic Division Setup**

For the polynomial  
$$
f(x)=x^4-16,
$$  
write it in full form (including all degrees):  
$$
f(x)=1x^4+0x^3+0x^2+0x-16.
$$  
The coefficients are:  
$$
1,\quad0,\quad0,\quad0,\quad-16.
$$

---

**1. Evaluating $$ f(3) $$ using Synthetic Division**

We set up synthetic division with $$ x=3 $$:

- Write the coefficients:  
  $$
  \begin{array}{cccccc}
  \text{Root: }3 & | & 1 & 0 & 0 & 0 & -16 \
  \hline
  & & 1 &  &  &  &  \
  \end{array}
  $$

- **Step 1:** Bring down the first coefficient: $$1$$.

- **Step 2:** Multiply $$1$$ by $$3$$ and write the result under the next coefficient: $$1\times3=3$$.  
  Now, add $$0+3=3$$.

- **Step 3:** Multiply the new value $$3$$ by $$3$$: $$3\times3=9$$.  
  Then, add: $$0+9=9$$.

- **Step 4:** Multiply $$9$$ by $$3$$: $$9\times3=27$$.  
  Add: $$0+27=27$$.

- **Step 5:** Multiply $$27$$ by $$3$$: $$27\times3=81$$.  
  Finally, add: $$-16+81=65$$.

The bottom row represents the coefficients of the depressed polynomial with the last number as the remainder, which is $$f(3)$$.

Thus,  
$$
f(3)=65.
$$

---

**2. Evaluating $$ f(-3) $$ using Synthetic Division**

Use the same coefficients and set $$ x=-3 $$:

- Coefficients:  
  $$
  1,\quad 0,\quad 0,\quad 0,\quad -16.
  $$

- **Step 1:** Bring down the first coefficient: $$1$$.

- **Step 2:** Multiply $$1$$ by $$-3$$: $$1\times(-3)=-3$$.  
  Add: $$0+(-3)=-3$$.

- **Step 3:** Multiply $$-3$$ by $$-3$$: $$-3\times(-3)=9$$.  
  Add: $$0+9=9$$.

- **Step 4:** Multiply $$9$$ by $$-3$$: $$9\times(-3)=-27$$.  
  Add: $$0+(-27)=-27$$.

- **Step 5:** Multiply $$-27$$ by $$-3$$: $$-27\times(-3)=81$$.  
  Add: $$-16+81=65$$.

Thus,  
$$
f(-3)=65.
$$

---

**3. Evaluating $$ f(1-\sqrt{6}) $$**

We now evaluate  
$$
f(1-\sqrt{6}) = (1-\sqrt{6})^4 - 16.
$$
While synthetic division can be used with irrational numbers, it is often simpler to compute using the binomial expansion.

Use the formula for $$(a-b)^4$$ with $$ a=1 $$ and $$ b=\sqrt{6} $$:
$$
(1-\sqrt{6})^4 = 1^4 - 4(1^3)(\sqrt{6}) + 6(1^2)(\sqrt{6})^2 - 4(1)(\sqrt{6})^3 + (\sqrt{6})^4.
$$

Now, compute each term:

1. $$1^4 = 1.$$
2. $$- 4(1^3)(\sqrt{6}) = -4\sqrt{6}.$$
3. $$6(1^2)(\sqrt{6})^2 = 6 \times 6 = 36,$$ since $$(\sqrt{6})^2=6.$$
4. $$- 4(1)(\sqrt{6})^3 = -4(6\sqrt{6}) = -24\sqrt{6},$$ because $$(\sqrt{6})^3=6\sqrt{6}.$$
5. $$(\sqrt{6})^4 = 36.$$

Combine the terms:
$$
(1-\sqrt{6})^4 = 1 - 4\sqrt{6} + 36 - 24\sqrt{6} + 36.
$$
Group similar terms:
- Constants: $$1 + 36 + 36 = 73.$$
- Radical terms: $$-4\sqrt{6} - 24\sqrt{6} = -28\sqrt{6}.$$

Thus,
$$
(1-\sqrt{6})^4 = 73 - 28\sqrt{6}.
$$
Subtract 16:
$$
f(1-\sqrt{6}) = (73 - 28\sqrt{6}) - 16 = 57 - 28\sqrt{6}.
$$

---

**Final Answers**

$$
\begin{aligned}
f(3) &= 65, \
f(-3) &= 65, \
f(1-\sqrt{6}) &= 57-28\sqrt{6}.
\end{aligned}
$$

So, the answer for  
$$
f(3)=\boxed{65}.
$$
$$ f(3) = 65 $$

Use synthetic division to find the function values.
find , and .
(Type an exact answer, using radicals as needed.)

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Use synthetic division to find the function values. find , and . (Type an exact answer, using radicals as needed.)