For the function $$ f(x)=x^{4}-3 x^{3}+x^{2}+4 x+1449 $$ use long division to determine whether $$ x+7 $$ is a factor of $$ f(x) $$.

Question

UpStudy AI · Accepted Answer

1. Write the function and the divisor:
   $$
   f(x)=x^{4}-3x^{3}+x^{2}+4x+1449, \quad \text{dividing by } (x+7).
   $$

2. Begin the long division. First, divide the leading term $$x^4$$ by $$x$$ to obtain:
   $$
   x^4 \div x = x^3.
   $$
   Multiply $$x^3$$ by $$(x+7)$$:
   $$
   x^3(x+7)=x^4+7x^3.
   $$
   Subtract this product from $$f(x)$$:
   $$
   \begin{array}{rcl}
   \quad & x^4-3x^3+x^2+4x+1449\
   - & \Bigl(x^4+7x^3\Bigr)\ \hline
   & -10x^3+x^2+4x+1449.
   \end{array}
   $$

3. Divide the new leading term $$-10x^3$$ by $$x$$ to get:
   $$
   -10x^3 \div x = -10x^2.
   $$
   Multiply $$-10x^2$$ by $$(x+7)$$:
   $$
   -10x^2(x+7)=-10x^3-70x^2.
   $$
   Subtract:
   $$
   \begin{array}{rcl}
   -10x^3+x^2+4x+1449\
   -\Bigl(-10x^3-70x^2\Bigr)\ \hline
   \quad\; 71x^2+4x+1449.
   \end{array}
   $$

4. Next, divide the new leading term $$71x^2$$ by $$x$$:
   $$
   71x^2 \div x = 71x.
   $$
   Multiply $$71x$$ by $$(x+7)$$:
   $$
   71x(x+7)=71x^2+497x.
   $$
   Subtract:
   $$
   \begin{array}{rcl}
   71x^2+4x+1449\
   -\Bigl(71x^2+497x\Bigr)\ \hline
   -493x+1449.
   \end{array}
   $$

5. Finally, divide $$-493x$$ by $$x$$:
   $$
   -493x \div x = -493.
   $$
   Multiply $$-493$$ by $$(x+7)$$:
   $$
   -493(x+7)=-493x-3451.
   $$
   Subtract:
   $$
   \begin{array}{rcl}
   -493x+1449\
   -\Bigl(-493x-3451\Bigr)\ \hline
   1449+3451=4900.
   \end{array}
   $$

6. The remainder of the division is $$4900$$.

Since the remainder is not zero, $$(x+7)$$ is not a factor of $$f(x)$$.
$$ x+7 $$ is not a factor of $$ f(x) $$.

For the function

use long division to determine whether is a factor of .

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions

For the function use long division to determine whether is a factor of .