Math / Algebra

QuestionQuestion Show Examples
Use the long division method to find the result when $3 x^{3}+16 x^{2}+10 x+25$ is divided $x+5$ .
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

What is this asking? We need to divide a polynomial, $3x^3 + 16x^2 + 10x + 25$ , by another polynomial, $x + 5$ , using long division and find the result. Watch out! Keep track of the signs and remember to multiply the divisor by the entire quotient term at each step.
Also, don't forget to write the remainder if there is one!

STEP 2

1. Set up the long division.
2. Divide the leading terms.
3. Multiply and subtract.
4. Repeat.

STEP 3

Alright, let's set up our long division problem!
We put the dividend, $3x^3 + 16x^2 + 10x + 25$ , inside the division symbol and the divisor, $x + 5$ , outside.
$\begin{array}{ccccc} x+5 & 3x^3 & +16x^2 & +10x & +25 \\ \end{array}$

STEP 4

Now, we divide the leading term of the dividend, $3x^3$ , by the leading term of the divisor, $x$ .
What do we get? $3x^3$ divided by $x$ is $3x^2$ !
We write this above the division symbol.
$\begin{array}{c ccc} x+5 & 3x^3 & +16x^2 & +10x+25 \\ \end{array}$

STEP 5

Next, we multiply our $3x^2$ by the entire divisor, $x + 5$ , which gives us $3x^3 + 15x^2$ .
We write this below the dividend and subtract it.
Remember to distribute that negative sign!
\begin{array}{c c c c} x+5 & 3x^3 & +16x^2 & +10x & +25 \\ & 3x^3 & +15x^2 & & \\ \cline{2-3} & 0 & x^2 & +10x & +25 \\ \end{array}

STEP 6

Now, we repeat the process!
Divide the leading term of the remaining polynomial, $x^2$ , by the leading term of the divisor, $x$ .
This gives us $x$ .
Write that above the division symbol next to the $3x^2$ .

STEP 7

Multiply $x$ by the divisor, $x + 5$ , to get $x^2 + 5x$ .
Subtract this from $x^2 + 10x + 25$ .
$\begin{array}{c cc cc} & 3x^2 & +x & & \\ x+5 & 3x^3 & +16x^2 & +10x & +25 \\ & 3x^3 & +15x^2 & & \\ & & x^2 & +10x & +25\\ & & x^2 & +5x & \\ & & & 5x & +25\\ \end{array}$

STEP 8

One last time!
Divide $5x$ by $x$ to get 5.
Multiply 5 by $x + 5$ to get $5x + 25$ .
Subtract this from $5x + 25$ to get 0!
$\begin{array}{cccc} 3x^2 & +x & +5 & \\ \hline 3x^3 & +16x^2 & +10x & +25 \\ 3x^3 & +15x^2 & & \\ \hline 0 & x^2 & +10x & +25\\ x^2 & +5x & & \\ \hline 0 & 5x & +25\\ 5x & +25\\ \hline 0 & 0\\ \end{array}$

STEP 9

The result of the division is $3x^2 + x + 5$ .

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QuestionQuestion Show Examples
Use the long division method to find the result when $3 x^{3}+16 x^{2}+10 x+25$ is divided $x+5$ .
Answer Attempt 1 out of 2 $\square$ Submit Answer

Studdy Solution

STEP 1

STEP 2

1. Set up the long division.
2. Divide the leading terms.
3. Multiply and subtract.
4. Repeat.

STEP 3

Alright, let's set up our long division problem!
We put the dividend, $3x^3 + 16x^2 + 10x + 25$ , inside the division symbol and the divisor, $x + 5$ , outside.
$\begin{array}{ccccc} x+5 & 3x^3 & +16x^2 & +10x & +25 \\ \end{array}$

STEP 4

Now, we divide the leading term of the dividend, $3x^3$ , by the leading term of the divisor, $x$ .
What do we get? $3x^3$ divided by $x$ is $3x^2$ !
We write this above the division symbol.
$\begin{array}{c ccc} x+5 & 3x^3 & +16x^2 & +10x+25 \\ \end{array}$

STEP 5

STEP 6

Now, we repeat the process!
Divide the leading term of the remaining polynomial, $x^2$ , by the leading term of the divisor, $x$ .
This gives us $x$ .
Write that above the division symbol next to the $3x^2$ .

STEP 7

Multiply $x$ by the divisor, $x + 5$ , to get $x^2 + 5x$ .
Subtract this from $x^2 + 10x + 25$ .
$\begin{array}{c cc cc} & 3x^2 & +x & & \\ x+5 & 3x^3 & +16x^2 & +10x & +25 \\ & 3x^3 & +15x^2 & & \\ & & x^2 & +10x & +25\\ & & x^2 & +5x & \\ & & & 5x & +25\\ \end{array}$

STEP 8

STEP 9

The result of the division is $3x^2 + x + 5$ .

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QuestionQuestion Show ExamplesUse the long division method to find the result when 3x3+16x2+10x+253 x^{3}+16 x^{2}+10 x+253x3+16x2+10x+25 is divided x+5x+5x+5.Answer Attempt 1 out of 2 □\square□ Submit Answer

Studdy Solution

STEP 1

STEP 2

1. Set up the long division.2. Divide the leading terms.3. Multiply and subtract.4. Repeat.

STEP 3

STEP 4

STEP 5

STEP 6

Now, we **repeat** the process!Divide the leading term of the remaining polynomial, x2x^2x2, by the leading term of the divisor, xxx.This gives us xxx.Write that above the division symbol next to the 3x23x^23x2.

STEP 7

STEP 8

STEP 9

The result of the division is 3x2+x+53x^2 + x + 53x2+x+5.

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QuestionQuestion Show Examples
Use the long division method to find the result when $3 x^{3}+16 x^{2}+10 x+25$ is divided $x+5$ .
Answer Attempt 1 out of 2 $\square$ Submit Answer

1. Set up the long division.
2. Divide the leading terms.
3. Multiply and subtract.
4. Repeat.

Now, we repeat the process!
Divide the leading term of the remaining polynomial, $x^2$ , by the leading term of the divisor, $x$ .
This gives us $x$ .
Write that above the division symbol next to the $3x^2$ .

The result of the division is $3x^2 + x + 5$ .