Use synthetic division to divide the polynomials. $$ \left(6 x^{2}-19 x+3\right) \div(x-3)=\square $$

Question

UpStudy AI · Accepted Answer

1. Write the polynomial $$6x^2 - 19x + 3$$ in descending order of degree and list the coefficients: $$6$$, $$-19$$, and $$3$$.

2. Since we are dividing by $$x-3$$, use $$3$$ as the synthetic divisor.

3. Set up the synthetic division:

$$
   \begin{array}{r|ccc}
   3 & 6 & -19 & 3 \
     &   &     &   \
     &   &     &   \
   \end{array}
   $$

4. Bring down the leading coefficient $$6$$:

$$
   \begin{array}{r|ccc}
   3 & 6 & -19 & 3 \
     &   &     &   \
     & 6 &     &   \
   \end{array}
   $$

5. Multiply $$6$$ by the divisor $$3$$ to get $$18$$. Write $$18$$ under $$-19$$ and add:

$$
   \begin{array}{r|ccc}
   3 & 6 & -19 & 3 \
     &   & 18  &   \
     & 6 & -1  &   \
   \end{array}
   $$

So, $$-19 + 18 = -1$$.

6. Multiply $$-1$$ by $$3$$ to get $$-3$$. Write $$-3$$ under $$3$$ and add:

$$
   \begin{array}{r|ccc}
   3 & 6 & -19 & 3 \
     &   & 18  & -3 \
     & 6 & -1  & 0 \
   \end{array}
   $$

Here, $$3 + (-3) = 0$$.

7. The bottom row (except the last term) gives the coefficients of the quotient:
   
   Quotient: $$6x - 1$$
   
   Remainder: $$0$$.

Thus, the result of the division is:

$$
\boxed{6x - 1}
$$
The result of dividing $$6x^2 - 19x + 3$$ by $$x - 3$$ is $$6x - 1$$.

Use synthetic division to divide the polynomials. \( \left(6 x^{2}-19 x+3\right) \div(x-3)=\square \)