How do you divide #(x^2+7x-5)div(x-2)# using long division?

Question

Christian Antunez · Accepted Answer

Alternative format of calculation

#x+9+13/(x-2)# #" "x^2+7x-5# #color(red)(x)(x-2)->" "ul(x^2-2x) larr" subtract"# #" "0+9x-5# #color(red)(9)(x-2)->" "ul(9x-18)larr" subtract" # #" "color(red)(0+13 larr" remainder")# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #(x^2=7x-5)-:(x-2)" "=" "color(red)(x+9+)(color(red)(13))/(x-2)#

Caleb Alexander · Answer

#(x^2+7x-5)div(x-2) = (x+9) " rem "13#
OR
#(x^2+7x-5)div(x-2) = (x+9) +13/(x-2)#
OR
#(x^2+7x-5) = (x-2)(x+9) +13# Algebraic long division follows the same method as arithmetic long division... #("dividend")/("divisor") = "quotient"# Step 1. Write the dividend in the 'box' making sure that the indices are in descending powers of x. Step 2. Divide the first term in divisor into the term in the dividend with the highest index. Write the answer at the top, Step 3. Multiply by BOTH terms of the divisor at the side Step 4. Subtract Step 5. Bring down the next term Repeat steps 2 to 5 #color(white)(xxxxxxxxxxxx)color(red)(x )color(blue)( + 9)" rem " 13# #color(white)(xxx)x-2 |bar( x^2 +7x -5)" "larr x^2divx = color(red)(x)# #color(white)(xxxxxx)ul(color(red)(-(x^2-2x)))color(white)(.)darr" "larr# subtract (change signs) #color(white)(xxxxxxxxxxxx) 9x-5""larr# bring down the -5, #9x div x = color(blue)(9)# #color(white)(xxxxxxxxxx)ul(color(blue)(-(9x-18))" "larr# subtract (change signs) #color(white)(xxxxxxxxx.xxxxxx)13 " "larr#remainder #(x^2+7x-5)div(x-2) = (x+9) " rem "13# This can also be written as #(x^2+7x-5)div(x-2) = (x+9) +13/(x-2)# Or#" "(x^2+7x-5) = (x-2)(x+9) +13#

Anna Allen · Answer

undefined To divide $ (x^2 + 7x - 5) $ by $ (x - 2) $ using long division, follow these steps:

1. Divide the first term of the dividend by the first term of the divisor and write the result above the horizontal division bar.
2. Multiply the divisor by the result obtained in step 1, and write the result below the dividend.
3. Subtract the result obtained in step 2 from the dividend, and bring down the next term from the dividend.
4. Repeat steps 1-3 until you have no more terms to bring down or until the degree of the remainder is less than the degree of the divisor.

Here are the steps performed for $ (x^2 + 7x - 5) \div (x - 2) $:

Step 1:
$$ \begin{array}{r|ll}
x & x & + 9 \
\hline
x - 2 & x^2 & + 7x & - 5 \
\end{array} $$

Step 2:
$$ \begin{array}{r|ll}
x & x & + 9 \
\hline
x - 2 & x^2 & + 7x & - 5 \
& \underline{x^2} & - 2x & \
& & 9x & \
\end{array} $$

Step 3:
$$ \begin{array}{r|ll}
x & x & + 9 \
\hline
x - 2 & x^2 & + 7x & - 5 \
& \underline{x^2} & - 2x & \
& & 9x & - 5 \
& & \underline{9x} & - 18 \
& & & 13 \
\end{array} $$

Therefore, $ (x^2 + 7x - 5) \div (x - 2) = x + 9 + \frac{13}{x - 2} $.