Use the long division method to find the result when 3, x, cubed, minus, 20, x, squared, plus, 29, x, minus, 263x

Question

3
−20x
2
+29x−26 is divided by x, minus, 5x−5. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+
b(x)
r(x)
​
.

Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, minus, 20, x, squared, minus, 27, x, plus, 19
4x
3
−20x
2
−27x+19 is divided by
x, minus, 6
x−6. If there is a remainder, express the result in the form
q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fraction
q(x)+
b(x)
r(x)
​
.
Use Long Division:
Use Long Division:

4, x, squared
4x
2
plus, 4, x
+4x
minus, 3
−3

x
x
minus, 6
−6
right parenthesis
)
4, x, cubed
4x
3
minus, 20, x, squared
−20x
2
minus, 27, x
−27x
plus, 19
+19

start superscript, minus, end superscript, 4, x, cubed
−
4x
3
minus, start superscript, plus, end superscript, 24, x, squared
−
+
24x
2

4, x, squared
4x
2
minus, 27, x
−27x
plus, 19
+19

start superscript, minus, end superscript, 4, x, squared
−
4x
2
minus, start superscript, plus, end superscript, 24, x
−
+
24x

minus, 3, x
−3x
plus, 19
+19

minus, start superscript, plus, end superscript, 3, x
−
+
3x
plus, start superscript, minus, end superscript, 18
+
−
18

1
1

Answer: , 4, x, squared, plus, 4, x, minus, 3, plus, start fraction, 1, divided by, x, minus, 6, end fraction
Answer: 4x
2
+4x−3+
x−6
1
​
Form: Quotient plus , start fraction, Remainder, divided by, Divisor, end fraction
Form: Quotient +
Divisor
Remainder
​

Answer 1

To perform long division on the polynomial \( 3x^3 - 20x^2 + 29x - 263 \) divided by \( x - 5 \), we will follow the long division steps meticulously.

Step 1: Set up the division

\[ \begin{array}{r|l} x - 5 & 3x^3 - 20x^2 + 29x - 263 \end{array} \]

Step 2: Divide the leading terms

Divide \( 3x^3 \) by \( x \):

\[ \frac{3x^3}{x} = 3x^2 \]

Step 3: Multiply and subtract

Multiply \( 3x^2 \) by \( (x - 5) \):

\[ 3x^2(x - 5) = 3x^3 - 15x^2 \]

Now, subtract this from the original polynomial:

\[ \begin{array}{r} 3x^3 - 20x^2 + 29x - 263 \ -(3x^3 - 15x^2) \ \hline 0 - (20x^2 - 15x^2) + 29x - 263 \ = -5x^2 + 29x - 263 \end{array} \]

Step 4: Repeat the process

Now divide \( -5x^2 \) by \( x \):

\[ \frac{-5x^2}{x} = -5x \]

Multiply \( -5x \) by \( (x - 5) \):

\[ -5x(x - 5) = -5x^2 + 25x \]

Subtract this from the current polynomial:

\[ \begin{array}{r} -5x^2 + 29x - 263 \ -(-5x^2 + 25x) \ \hline 0 + (29x - 25x) - 263 \ = 4x - 263 \end{array} \]

Step 5: Repeat again

Now divide \( 4x \) by \( x \):

\[ \frac{4x}{x} = 4 \]

Multiply \( 4 \) by \( (x - 5) \):

\[ 4(x - 5) = 4x - 20 \]

Subtract this from the current polynomial:

\[ \begin{array}{r} 4x - 263 \ -(4x - 20) \ \hline 0 - (-263 + 20) \ = -243 \end{array} \]

Step 6: Conclusion

Now we have completed the division. The quotient \( q(x) \) is:

\[ q(x) = 3x^2 - 5x + 4 \]

The remainder \( r(x) \) is:

\[ r(x) = -243 \]

The result can be expressed in the required form:

\[ q(x) + \frac{r(x)}{b(x)} = 3x^2 - 5x + 4 + \frac{-243}{x - 5} \]

Final Answer:

\[ 3x^2 - 5x + 4 - \frac{243}{x - 5} \]