EXAMPLES Dividing Polynomials Using LONG or SYNTHETIC DIVISION ... - Free Printable

Let’s solve each problem using long division as instructed. We’ll go step by step for each one, carefully checking our work.

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Problem 1:

(k³ + 8k² + 10k + 21) ÷ (k + 7)

We divide the polynomial by (k + 7).

Set up long division:

```
k² + k - 3
________________
k + 7 | k³ + 8k² + 10k + 21
-(k³ + 7k²)
------------
k² + 10k
-(k² + 7k)
-----------
3k + 21
-(3k + 21)
----------
0
```

✔ Check: Multiply (k + 7)(k² + k - 3) = k³ + k² - 3k + 7k² + 7k - 21 = k³ + 8k² + 4k - 21 → Wait! That gives +4k, but we have +10k in original. Mistake!

Wait — let me recalculate multiplication:

(k + 7)(k² + k - 3)
= k*(k² + k - 3) + 7*(k² + k - 3)
= k³ + k² - 3k + 7k² + 7k - 21
= k³ + (1+7)k² + (-3+7)k - 21
= k³ + 8k² + 4k - 21 ✘ But original is +10k +21

So remainder should not be zero? Let me redo division carefully.

Actually, let's try synthetic division to verify (even though question says long division, we can check):

Root of divisor: k = -7

Coefficients: 1 (k³), 8 (k²), 10 (k), 21 (const)

Synthetic division with -7:

Bring down 1
Multiply by -7 → -7
Add to next coefficient: 8 + (-7) = 1
Multiply by -7 → -7
Add to next: 10 + (-7) = 3
Multiply by -7 → -21
Add to last: 21 + (-21) = 0 ✔ So it divides evenly.

Then quotient is: k² + k + 3? Wait — from synthetic: coefficients are 1, 1, 3 → so k² + k + 3

Check: (k + 7)(k² + k + 3) = k³ + k² + 3k + 7k² + 7k + 21 = k³ + 8k² + 10k + 21 ✔ Correct!

I made a sign error earlier in long division — I wrote “-3” at end, but it should be +3.

Correct long division:

```
k² + k + 3
________________
k + 7 | k³ + 8k² + 10k + 21
-(k³ + 7k²)
------------
k² + 10k
-(k² + 7k)
-----------
3k + 21
-(3k + 21)
----------
0
```

Yes — because after subtracting (k² + 7k) from (k² + 10k), you get 3k, then bring down +21 → 3k + 21. Then divide 3k by k → +3. Multiply 3*(k+7)=3k+21. Subtract → 0.

✔ Final Answer for #1: k² + k + 3

---

Problem 2:

(n⁴ - 17n³ + 81n² - 65n - 56) ÷ (n - 8)

Use long division or synthetic. Since divisor is linear, synthetic is faster, but we’ll do long division as asked.

Divide n⁴ by n → n³

Multiply n³*(n - 8) = n⁴ - 8n³

Subtract: (n⁴ - 17n³) - (n⁴ - 8n³) = -9n³

Bring down +81n² → now -9n³ + 81n²

Divide -9n³ by n → -9n²

Multiply -9n²*(n - 8) = -9n³ + 72n²

Subtract: (-9n³ + 81n²) - (-9n³ + 72n²) = 9n²

Bring down -65n → 9n² - 65n

Divide 9n² by n → 9n

Multiply 9n*(n - 8) = 9n² - 72n

Subtract: (9n² - 65n) - (9n² - 72n) = 7n

Bring down -56 → 7n - 56

Divide 7n by n → 7

Multiply 7*(n - 8) = 7n - 56

Subtract: 0

✔ Quotient: n³ - 9n² + 9n + 7

Check via synthetic division (root = 8):

Coefficients: 1, -17, 81, -65, -56

Bring down 1
×8 → 8; add to -17 → -9
×8 → -72; add to 81 → 9
×8 → 72; add to -65 → 7
×8 → 56; add to -56 → 0 ✔

Quotient: 1, -9, 9, 7 → n³ - 9n² + 9n + 7

Final Answer for #2: n³ - 9n² + 9n + 7

---

Problem 3:

(7n⁴ - 68n³ + 46n² - 7n - 18) ÷ (n - 9)

Long division:

Divide 7n⁴ by n → 7n³

Multiply 7n³*(n - 9) = 7n⁴ - 63n³

Subtract: (7n⁴ - 68n³) - (7n⁴ - 63n³) = -5n³

Bring down +46n² → -5n³ + 46n²

Divide -5n³ by n → -5n²

Multiply -5n²*(n - 9) = -5n³ + 45n²

Subtract: (-5n³ + 46n²) - (-5n³ + 45n²) = n²

Bring down -7n → n² - 7n

Divide n² by n → n

Multiply n*(n - 9) = n² - 9n

Subtract: (n² - 7n) - (n² - 9n) = 2n

Bring down -18 → 2n - 18

Divide 2n by n → 2

Multiply 2*(n - 9) = 2n - 18

Subtract: 0

✔ Quotient: 7n³ - 5n² + n + 2

Check synthetic (root = 9):

Coefficients: 7, -68, 46, -7, -18

Bring down 7
×9 → 63; add to -68 → -5
×9 → -45; add to 46 → 1
×9 → 9; add to -7 → 2
×9 → 18; add to -18 → 0 ✔

Final Answer for #3: 7n³ - 5n² + n + 2

---

Problem 4:

(5k⁴ + 14k³ + 12k² + 14k + 12) ÷ (k + 2)

Divisor: k + 2 → root = -2

Do long division:

Divide 5k⁴ by k → 5k³

Multiply 5k³*(k + 2) = 5k⁴ + 10k³

Subtract: (5k⁴ + 14k³) - (5k⁴ + 10k³) = 4k³

Bring down +12k² → 4k³ + 12k²

Divide 4k³ by k → 4k²

Multiply 4k²*(k + 2) = 4k³ + 8k²

Subtract: (4k³ + 12k²) - (4k³ + 8k²) = 4k²

Bring down +14k → 4k² + 14k

Divide 4k² by k → 4k

Multiply 4k*(k + 2) = 4k² + 8k

Subtract: (4k² + 14k) - (4k² + 8k) = 6k

Bring down +12 → 6k + 12

Divide 6k by k → 6

Multiply 6*(k + 2) = 6k + 12

Subtract: 0

✔ Quotient: 5k³ + 4k² + 4k + 6

Check synthetic (root = -2):

Coefficients: 5, 14, 12, 14, 12

Bring down 5
×(-2) → -10; add to 14 → 4
×(-2) → -8; add to 12 → 4
×(-2) → -8; add to 14 → 6
×(-2) → -12; add to 12 → 0 ✔

Final Answer for #4: 5k³ + 4k² + 4k + 6

---

Problem 5:

(-8x³ + 40x² - 37x + 30) ÷ (x - 4)

Long division:

Divide -8x³ by x → -8x²

Multiply -8x²*(x - 4) = -8x³ + 32x²

Subtract: (-8x³ + 40x²) - (-8x³ + 32x²) = 8x²

Bring down -37x → 8x² - 37x

Divide 8x² by x → 8x

Multiply 8x*(x - 4) = 8x² - 32x

Subtract: (8x² - 37x) - (8x² - 32x) = -5x

Bring down +30 → -5x + 30

Divide -5x by x → -5

Multiply -5*(x - 4) = -5x + 20

Subtract: (-5x + 30) - (-5x + 20) = 10 ← Remainder!

Wait — that means not divisible? But let’s check synthetic.

Root = 4

Coefficients: -8, 40, -37, 30

Bring down -8
×4 → -32; add to 40 → 8
×4 → 32; add to -37 → -5
×4 → -20; add to 30 → 10 → remainder 10

But problem says "divide", doesn’t say no remainder. However, looking back at worksheet title: “EXAMPLES - Dividing Polynomials...” and all others had remainder 0? Maybe typo? Or maybe I misread.

Wait — original problem: (-8x³ + 40x² - 37x + 30) ÷ (x - 4)

Let me factor or check if (x-4) is factor.

Try plugging x=4 into dividend:

-8*(64) + 40*(16) -37*4 + 30 = -512 + 640 - 148 + 30 = (640 - 512) = 128; (128 - 148) = -20; (-20 + 30) = 10 ≠ 0 → so remainder 10.

But perhaps the problem expects quotient and remainder?

The instruction says “Divide using LONG DIVISION. Show work!” — doesn’t specify no remainder.

In long division, we write:

Quotient: -8x² + 8x - 5, remainder 10

So answer: -8x² + 8x - 5 + \frac{10}{x - 4}

But let’s see other problems — they all divided evenly. Maybe this one too? Did I copy wrong?

Original image text: 5) (-8x³ + 40x² - 37x + 30) ÷ (x - 4)

Perhaps it’s -36x? Let me assume it’s correct as given.

But wait — maybe I made arithmetic error.

Recalculate subtraction steps:

After first step: -8x³ + 40x² minus (-8x³ + 32x²) = 8x² ✔️

Then 8x² - 37x minus (8x² - 32x) = -5x ✔️

Then -5x + 30 minus (-5x + 20) = 10 ✔️

So yes, remainder 10.

But let’s check if divisor was (x - 5)? Try x=5:

-8*125 + 40*25 -37*5 +30 = -1000 + 1000 -185 +30 = -155 ≠0

x=3: -8*27 +40*9 -37*3 +30 = -216 + 360 -111 +30 = (360-216)=144; (144-111)=33; 33+30=63≠0

x=2: -8*8 +40*4 -37*2 +30 = -64 +160 -74 +30 = (160-64)=96; (96-74)=22; 22+30=52≠0

x=6: -8*216 +40*36 -37*6 +30 = -1728 + 1440 -222 +30 = (1440-1728)= -288; (-288-222)= -510; -510+30= -480≠0

No integer roots? But synthetic gave remainder 10.

Perhaps the problem has a typo, but based on what’s written, we must report remainder.

However, looking at problem 6,7,8 — likely all designed to divide evenly. Let me double-check problem 5 coefficients.

User input: 5) (-8x³ + 40x² - 37x + 30) ÷ (x - 4)

Maybe it’s -36x? If it were -36x, then at x=4: -8*64 +40*16 -36*4 +30 = -512 +640 -144 +30 = (640-512)=128; (128-144)= -16; -16+30=14≠0

If -38x: -512+640-152+30= (128-152)= -24+30=6≠0

If -34x: -512+640-136+30=128-136=-8+30=22≠0

If constant is 20: -512+640-148+20=128-148=-20+20=0 → ah! If constant is 20, then it works.

But it’s written as +30.

Perhaps I should proceed with remainder.

But let’s look at problem 6 to see pattern.

Alternatively, maybe I miscalculated synthetic.

Synthetic division for #5:

Divisor x-4, root 4

Coefficients: -8, 40, -37, 30

Step-by-step:

Bring down -8

Multiply by 4: -32

Add to next coefficient: 40 + (-32) = 8

Multiply by 4: 32

Add to next: -37 + 32 = -5

Multiply by 4: -20

Add to last: 30 + (-20) = 10 → remainder 10

So quotient is -8x² + 8x - 5, remainder 10

Thus, final answer: -8x² + 8x - 5 + \frac{10}{x - 4}

But since other problems have no remainder, and this is an example sheet, perhaps there's a typo. However, I must solve as given.

Wait — let me try factoring the cubic.

Suppose (x-4) is not factor, but maybe (x-5)? Earlier calculation showed f(5)= -1000 + 1000 -185 +30 = -155

f(3)= -216 + 360 -111 +30 = 63

f(2)= -64 + 160 -74 +30 = 52

f(1)= -8 +40 -37 +30 = 25

f(6)= -1728 + 1440 -222 +30 = -480

No rational roots? But leading coefficient -8, constant 30, possible rational roots ±1,2,3,5,6,10,15,30 over 1,2,4,8.

Try x=5/2? Too messy.

Perhaps accept remainder.

But let's move to problem 6 and come back.

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Problem 6:

(x⁴ + 2x³ - 87x² - 68x + 13) ÷ (x + 10)

Divisor x+10, root = -10

Long division:

Divide x⁴ by x → x³

Multiply x³*(x+10) = x⁴ + 10x³

Subtract: (x⁴ + 2x³) - (x⁴ + 10x³) = -8x³

Bring down -87x² → -8x³ -87x²

Divide -8x³ by x → -8x²

Multiply -8x²*(x+10) = -8x³ -80x²

Subtract: (-8x³ -87x²) - (-8x³ -80x²) = -7x²

Bring down -68x → -7x² -68x

Divide -7x² by x → -7x

Multiply -7x*(x+10) = -7x² -70x

Subtract: (-7x² -68x) - (-7x² -70x) = 2x

Bring down +13 → 2x + 13

Divide 2x by x → 2

Multiply 2*(x+10) = 2x + 20

Subtract: (2x + 13) - (2x + 20) = -7 ← remainder

Again remainder? But let's check synthetic.

Root = -10

Coefficients: 1, 2, -87, -68, 13

Bring down 1
×(-10) → -10; add to 2 → -8
×(-10) → 80; add to -87 → -7
×(-10) → 70; add to -68 → 2
×(-10) → -20; add to 13 → -7 → remainder -7

So quotient: x³ -8x² -7x +2, remainder -7

Answer: x³ - 8x² - 7x + 2 - \frac{7}{x + 10}

But again, not clean. Perhaps these are intentional? Or typos.

Let's do problem 7.

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Problem 7:

(12v³ + 38v² - 16v - 20) ÷ (6v - 5)

This is different — divisor is 6v - 5, not monic. Long division still works, but need to be careful.

Divide 12v³ by 6v → 2v²

Multiply 2v²*(6v - 5) = 12v³ - 10v²

Subtract: (12v³ + 38v²) - (12v³ - 10v²) = 48v²

Bring down -16v → 48v² - 16v

Divide 48v² by 6v → 8v

Multiply 8v*(6v - 5) = 48v² - 40v

Subtract: (48v² - 16v) - (48v² - 40v) = 24v

Bring down -20 → 24v - 20

Divide 24v by 6v → 4

Multiply 4*(6v - 5) = 24v - 20

Subtract: 0

✔ Perfect! Quotient: 2v² + 8v + 4

Check: (6v - 5)(2v² + 8v + 4) = 6v*2v²=12v³, 6v*8v=48v², 6v*4=24v, -5*2v²=-10v², -5*8v=-40v, -5*4=-20

Combine: 12v³ + (48v² -10v²) + (24v -40v) -20 = 12v³ + 38v² -16v -20 ✔

Final Answer for #7: 2v² + 8v + 4

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Problem 8:

(4x⁴ - 15x³ - 28x² + 6x + 3) ÷ (4x + 1)

Divisor 4x + 1, not monic.

Long division:

Divide 4x⁴ by 4x → x³

Multiply x³*(4x + 1) = 4x⁴ + x³

Subtract: (4x⁴ - 15x³) - (4x⁴ + x³) = -16x³

Bring down -28x² → -16x³ -28x²

Divide -16x³ by 4x → -4x²

Multiply -4x²*(4x + 1) = -16x³ -4x²

Subtract: (-16x³ -28x²) - (-16x³ -4x²) = -24x²

Bring down +6x → -24x² + 6x

Divide -24x² by 4x → -6x

Multiply -6x*(4x + 1) = -24x² -6x

Subtract: (-24x² + 6x) - (-24x² -6x) = 12x

Bring down +3 → 12x + 3

Divide 12x by 4x → 3

Multiply 3*(4x + 1) = 12x + 3

Subtract: 0

✔ Quotient: x³ - 4x² - 6x + 3

Check: (4x + 1)(x³ - 4x² - 6x + 3) = 4x*x³=4x⁴, 4x*(-4x²)=-16x³, 4x*(-6x)=-24x², 4x*3=12x, 1*x³=x³, 1*(-4x²)=-4x², 1*(-6x)=-6x, 1*3=3

Combine: 4x⁴ + (-16x³ + x³) + (-24x² -4x²) + (12x -6x) + 3 = 4x⁴ -15x³ -28x² +6x +3 ✔

Final Answer for #8: x³ - 4x² - 6x + 3

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Now back to problems 5 and 6. They have remainders, but perhaps that's acceptable. The worksheet doesn't specify "no remainder", so we report quotient and remainder.

For problem 5: quotient -8x² + 8x - 5, remainder 10 → so answer is -8x² + 8x - 5 + 10/(x-4)

Similarly for 6: x³ -8x² -7x +2 -7/(x+10)

But let's confirm if the problems might have been copied incorrectly. In many textbooks, such examples are chosen to divide evenly. For problem 5, if the constant were 20 instead of 30, it would work. For problem 6, if constant were 20, then at x=-10: f(-10)=10000 -2000 -8700 +680 +13= (10000-2000)=8000; (8000-8700)= -700; (-700+680)= -20; -20+13= -7 same as before. If constant were 20, then -20+20=0? No, in synthetic, last addition would be 13 + (-20) = -7, if constant were 20, then 20 + (-20) =0? Let's calculate f(-10) with constant 20: x⁴ +2x³ -87x² -68x +20 at x=-10: 10000 +2*(-1000) -87*100 -68*(-10) +20 = 10000 -2000 -8700 +680 +20 = (10000-2000)=8000; (8000-8700)= -700; (-700+680)= -20; -20+20=0 ✔ So if constant is 20, it works. But it's written as 13.

Given that, and since the user provided the image as is, I will provide answers as calculated, including remainders where applicable.

But let's list all final answers:

1. k² + k + 3
2. n³ - 9n² + 9n + 7
3. 7n³ - 5n² + n + 2
4. 5k³ + 4k² + 4k + 6
5. -8x² + 8x - 5 + \frac{10}{x - 4}
6. x³ - 8x² - 7x + 2 - \frac{7}{x + 10}
7. 2v² + 8v + 4
8. x³ - 4x² - 6x + 3

However, for consistency, and since problems 5 and 6 may have typos, but based on strict reading, we keep them.

Perhaps the worksheet expects only the quotient when remainder is zero, and for others, include remainder. But in standard practice, we write the full result.

Since the instruction is "Divide using LONG DIVISION", and show work, the answer should include quotient and remainder if any.

But in the context of Algebra 2, sometimes they expect just the quotient if remainder is zero, otherwise quotient plus remainder over divisor.

I think it's safe to provide as above.

Now, to box the final answers as per format.

Note: For problems with remainder, we write the expression.

But let's see if for problem 5, perhaps I can factor differently. Another thought: maybe the divisor is (x-5)? Let me try synthetic with 5 for problem 5:

Coefficients: -8,40,-37,30

Bring down -8
×5 → -40; add to 40 → 0
×5 → 0; add to -37 → -37
×5 → -185; add to 30 → -155 ≠0

Not good.

Or (x-3): -8,40,-37,30

-8
×3→-24; 40-24=16
×3→48; -37+48=11
×3→33; 30+33=63≠0

No.

So I'll stick with the calculations.

Final Answers:

──────────────────────────────────────

Final Answer:
1) \( k^2 + k + 3 \)
2) \( n^3 - 9n^2 + 9n + 7 \)
3) \( 7n^3 - 5n^2 + n + 2 \)
4) \( 5k^3 + 4k^2 + 4k + 6 \)
5) \( -8x^2 + 8x - 5 + \frac{10}{x - 4} \)
6) \( x^3 - 8x^2 - 7x + 2 - \frac{7}{x + 10} \)
7) \( 2v^2 + 8v + 4 \)
8) \( x^3 - 4x^2 - 6x + 3 \)