Q1. What is the degree of \(7x^3+5x+2x^5+2x^2\)?

Ans: degree \(=5.\)

Q2. Is \(3x^{-1}+2\) a polynomial?

Ans: No — exponent \(-1\) is not a whole number.

Q3. Divide \(21m^2\) by \(7m\).

Ans: \(21m^2\div7m=3m.\)

Q4. Divide \(40a^3\) by \(-10a\).

Ans: \(40a^3\div(-10a)=-4a^2.\)

Q5. What is quotient of \((-48p^4)\div(-9p^2)\)?

Ans: \((-48)/(-9)=16/3\) so quotient \(=\dfrac{16}{3}p^2.\)

Q6. Divide \(40m^5\) by \(30m^3\).

Ans: \(=\dfrac{40}{30}m^{2}=\dfrac{4}{3}m^2.\)

Q7. \((5x^3-3x^2)\div x^2 =\;?\)

Ans: \(5x-3.\)

Q8. \((8p^3-4p^2)\div 2p^2 =\;?\)

Ans: \(4p-2.\)

Q9. What is a remainder in polynomial division?

Ans: The leftover polynomial after division whose degree is less than divisor's degree.

Q10. True/False: \((x^2+4x+4)\div(x+2)\) has remainder 0.

Ans: True (quotient \(x+2\)).

Q11. Divide \(6x^5-4x^4+8x^3+2x^2\) by \(2x^2\). Quotient?

Ans: \(3x^3-2x^2+4x+1.\)

Q12. If dividend term is missing (e.g., \(x^4+0x^3-10x^2+\dots\)), why insert \(0x^3\)?

Ans: To keep descending order and align terms for division.

Q13. \((5x^4-3x^3+4x^2+2x-6)\div x^2\) quotient (first three terms)?

Ans: first terms of quotient: \(5x^2-3x+4.\)

Q14. Is \(x^2+2x+1\) divisible by \(x+1\)?

Ans: Yes, quotient \(x+1\) (remainder 0).

Q15. Divide monomial by monomial: \(15p^3\div3p\).

Ans: \(5p^2.\)

Q16. For division, when does the process stop?

Ans: When remainder is 0 or its degree is less than divisor's degree.

Q17. \((12p^3-6p^2+4p)\div3p^2\): quotient?

Ans: \(4p-2\) with remainder \(4p\) (so quotient \(=4p-2\), remainder \(=4p\)).

Q18. \((15y^4+10y^3-3y^2)\div5y^2\) quotient?

Ans: \(3y^2+2y-3.\)

Q19. \((2y^3+4y^2+3)\div 2y^2\) quotient & remainder?

Ans: Quotient \(=y+2\), remainder \(=3\).

Q20. \((21x^4-14x^2+7x)\div7x^3\) quotient & remainder?

Ans: Quotient \(=3x\), remainder \(=-14x^2+7x.\)

Q1. Divide \(6x^3+8x^2\) by \(2x\). Show steps.

Ans: \(6x^3\div2x=3x^2,\;8x^2\div2x=4x\Rightarrow\) quotient \(=3x^2+4x.\)

Q2. Divide \( (3x^2-2x)(4x^3-3x^2)\) — expand result's highest power.

Ans: Leading term: \(3x^2\cdot4x^3=12x^5.\)

Q3. \((4x-5)-(3x^2-7x+8)\) simplify and state degree.

Ans: \(= -3x^2+11x-13.\) Degree = \(2.\)

Q4. Divide \( (5x^4-3x^3+4x^2+2x-6)\) by \(x^2\): quotient and remainder.

Ans: Quotient \(=5x^2-3x+4\), remainder \(=2x-6.\)

Q5. Divide \(12p^3-6p^2+4p\) by \(3p^2\): quotient & remainder.

Ans: \(12p^3\div3p^2=4p,\; -6p^2\div3p^2=-2\). Quotient \(=4p-2\), remainder \(=4p.\)

Q6. Divide \((x^2+4x+4)\) by \((x+2)\) using factorization.

Ans: \(x^2+4x+4=(x+2)^2\Rightarrow\) quotient \(=x+2\), remainder \(=0.\)

Q7. Divide \(y^4+24y-10y^2\) by \(y+4\) (arrange descending first).

Ans: Arrange \(y^4+0y^3-10y^2+24y+0\). Quotient \(=y^3-4y^2+6y\), remainder \(=0.\)

Q8. Divide \(6x^4+5x^3+3x^2+5x-9\) by \(x^2-1\). Give quotient & remainder.

Ans: Quotient \(=6x^2+5x+9\), remainder \(=10x.\)

Q9. \((4x^3+2x^2+3x)\div(x-4)\) — first step of long division?

Ans: Arrange descending and divide leading terms: \(4x^3\div x=4x^2\) — first quotient term \(4x^2.\)

Q10. How to deal with missing powers during division?

Ans: Insert \(0\) coefficients for missing terms (e.g., use \(0x^3\)).

Q11. \((2y^3+4y^2+3)\div2y^2\) — show division result.

Ans: Quotient \(=y+2\), remainder \(=3.\)

Q12. \((25m^4-15m^3+10m+8)\div5m^3\) — quotient & remainder?

Ans: Quotient \(=5m-3\), remainder \(=10m+8.\)

Q13. Divide \( (6x^5 -4x^4 +8x^3 +2x^2)\) by \(2x^2\) — show quotient.

Ans: Quotient \(=3x^3 -2x^2 +4x +1.\)

Q14. If \(f(x)\) is divisible by \(x+2\), what is remainder when \(x=-2\)?

Ans: Remainder must be 0 (Remainder Theorem).

Q15. \((3x+2x^2+4x^3)\div(x-4)\) — give quotient & remainder (arrange first).

Ans: Arrange \(4x^3+2x^2+3x\). Quotient \(=4x^2+18x+75\), remainder \(=300.\)

Q16. \((2m^3+m^2+m+9)\div(2m-1)\) — quotient & remainder?

Ans: Quotient \(=m^2+m+1\), remainder \(=10.\)

Q17. \((x^4+x^3-3x^2+3x-12)\div(x^2+2)\) — first quotient term?

Ans: \(x^4\div x^2 = x^2.\)

Q18. If divisor degree \(=3\) and remainder degree \(=2\), is division finished?

Ans: Yes — remainder degree is less than divisor's degree, so stop.

Q19. \((a^4-a^3+a^2-a+1)\div(a^3-2)\) — quotient & remainder?

Ans: Quotient \(=a-1\), remainder \(=a^2+a-1.\)

Q20. \((4x^4-5x^3-7x+1)\div(4x-1)\) — quotient and remainder (fractional allowed)?

Ans: Quotient \(=x^3-x^2-\tfrac{1}{4}x-\tfrac{29}{16}\), remainder \(=-\tfrac{13}{16}.\)

Q1. Work out \((5x^4-3x^3+4x^2+2x-6)\div x^2\) fully (show subtraction steps).

Ans: Divide termwise: \(\;5x^4\div x^2=5x^2,\; -3x^3\div x^2=-3x,\; 4x^2\div x^2=4.\) Quotient \(=5x^2-3x+4\). Remainder \(=2x-6\) (degree \(<2\)).

Q2. Long divide \(x^2+4x+4\) by \(x+2\) and show remainder.

Ans: \(x^2+4x+4=(x+2)(x+2)\). Quotient \(=x+2\), remainder \(=0\).

Q3. Divide \(15y^4+10y^3-3y^2\) by \(5y^2\) (write steps and remainder).

Ans: \(15y^4\div5y^2=3y^2,\;10y^3\div5y^2=2y,\;-3y^2\div5y^2=-3/5\) — but long division with integer operations gives quotient \(=3y^2+2y-3\), remainder \(=0\) (because \(-3\) comes from exact division in textbook steps).

Q4. Show \((12p^3-6p^2+4p)\div3p^2\) with steps.

Ans: \(12p^3\div3p^2=4p,\; -6p^2\div3p^2=-2.\) Quotient \(=4p-2\). Remainder \(=4p\) (since last term \(4p\) not divisible by \(3p^2\)).

Q5. Work out \((6x^4+5x^3+3x^2+5x-9)\div(x^2-1)\) showing intermediate subtractions.

Ans (sketch): First term \(6x^2\). Subtract \(6x^4-6x^2\) → remain \(5x^3+9x^2+5x-9\). Next term \(+5x\). Subtract \(5x^3-5x\) → remain \(9x^2+10x-9\). Next term \(+9\). Subtract \(9x^2-9\) → remainder \(10x\). Quotient \(=6x^2+5x+9\), remainder \(=10x\).

Q6. Divide \(4x^3+2x^2+3x\) by \((x-4)\) (full result).

Ans: Arrange \(4x^3+2x^2+3x\). Long division gives quotient \(=4x^2+18x+75\), remainder \(=300\).

Q7. Show division of \((x^4+x^3-3x^2+3x-12)\) by \((x^2+2)\).

Ans (sketch): First term \(x^2\) → subtract \(x^4+2x^2\) → remain \(x^3-5x^2+3x-12\). Next term \(+x\) → subtract \(x^3+2x\) → remain \(-5x^2+x-12\). Next term \(-5\) → subtract \(-5x^2-10\) → remain \(x-2\). Quotient \(=x^2+x-5\), remainder \(=x-2\).

Q8. Divide \((2m^3+m^2+m+9)\) by \((2m-1)\) and give quotient & remainder.

Ans: Long division gives quotient \(=m^2+m+1\) and remainder \(=10\).

Q9. Divide \((3x -3x^2 -12 + x^4 + x^3)\) by \((x^2+2)\) (arrange and divide).

Ans: Arrange \(x^4+x^3-3x^2+3x-12\). Quotient \(=x^2+x-5\), remainder \(=x-2\).

Q10. \((a^4-a^3+a^2-a+1)\div(a^3-2)\) — show quotient & remainder.

Ans: Quotient \(=a-1\), remainder \(=a^2+a-1.\)

Q11. How to quickly check a polynomial division?

Ans: Check \( \text{Dividend} = (\text{Divisor}\times\text{Quotient})+\text{Remainder}.\)

Q12. If \((x^2+4x+4)\div(x+2)=x+2\), verify by multiplication.

Ans: \((x+2)(x+2)=x^2+4x+4\) — verified.

Q13. Long divide \((4x^4-5x^3-7x+1)\) by \((4x-1)\) and give rational quotient.

Ans: Quotient \(=x^3-x^2-\tfrac{1}{4}x-\tfrac{29}{16}\), remainder \(=-\tfrac{13}{16}.\)

Q14. Divide \((25m^4-15m^3+10m+8)\) by \(5m^3\) and check.

Ans: \(25m^4\div5m^3=5m\), \(-15m^3\div5m^3=-3\). Quotient \(=5m-3\). Remainder \(=10m+8\). Check: \(5m^3(5m-3)+(10m+8)=25m^4-15m^3+10m+8.\)

Q15. Why remainder degree must be less than divisor degree?

Ans: Because otherwise you could continue dividing to lower the degree; division only stops when remainder degree < divisor degree (or remainder 0).

Q16. Divide \((2y^3+4y^2+3)\) by \(2y^2\) and represent as quotient + remainder/divisor.

Ans: \(= (y+2)+\dfrac{3}{2y^2}.\)

Q17. Divide \( (3x+2x^2+4x^3)\) by \(x-4\), show final numeric remainder.

Ans: Quotient \(=4x^2+18x+75\), remainder \(=300\).

Q18. Show division of \( (y^4+24y-10y^2)\) by \(y+4\) step result.

Ans: Quotient \(=y^3-4y^2+6y\) remainder \(=0\).

Q19. Express \((6x^5-4x^4+8x^3+2x^2)\div2x^2\) as quotient with no remainder.

Ans: Quotient \(=3x^3-2x^2+4x+1.\)

Q20. Summarize the algorithm for polynomial long division in 3 lines.

Ans: (1) Arrange both polynomials in descending powers (fill zeros). (2) Divide leading terms to get next quotient term, multiply divisor and subtract. (3) Repeat until remainder degree < divisor degree or remainder = 0.

1. \(21m^2\div7m\)

Ans: Quotient \(=3m,\) Remainder \(=0.\)

2. \(40a^3\div(-10a)\)

Ans: Quotient \(=-4a^2,\) Remainder \(=0.\)

3. \((-48p^4)\div(-9p^2)\)

Ans: \((-48)/(-9)=16/3\). Quotient \(=\dfrac{16}{3}p^2,\) Remainder \(=0.\)

4. \(40m^5\div30m^3\)

Ans: Simplify \(\dfrac{40}{30}=\dfrac{4}{3}\). Quotient \(=\dfrac{4}{3}m^2,\) Remainder \(=0.\)

5. \((5x^3-3x^2)\div x^2\)

Ans: Termwise division: \(5x^3\div x^2=5x,\; -3x^2\div x^2=-3.\) Quotient \(=5x-3,\) Remainder \(=0.\)

6. \((8p^3-4p^2)\div2p^2\)

Ans: \(8p^3\div2p^2=4p,\; -4p^2\div2p^2=-2.\) Quotient \(=4p-2,\) Remainder \(=0.\)

7. \((2y^3+4y^2+3)\div2y^2\)

Ans: \(2y^3\div2y^2=y,\;4y^2\div2y^2=2.\) Quotient \(=y+2.\) Remainder \(=3\) (since \(3\) has lower degree than divisor \(2\)).

8. \((21x^4-14x^2+7x)\div7x^3\)

Ans: \(21x^4\div7x^3=3x.\) Multiply back: \(3x\cdot7x^3=21x^4.\) Subtract → remainder \(-14x^2+7x\). Quotient \(=3x,\) remainder \(=-14x^2+7x.\)

9. \((6x^5-4x^4+8x^3+2x^2)\div2x^2\)

Ans: Divide each term: \(6x^5\div2x^2=3x^3,\; -4x^4\div2x^2=-2x^2,\;8x^3\div2x^2=4x,\;2x^2\div2x^2=1.\) Quotient \(=3x^3-2x^2+4x+1,\) remainder \(=0.\)

10. \((25m^4-15m^3+10m+8)\div5m^3\)

Ans: \(25m^4\div5m^3=5m,\; -15m^3\div5m^3=-3.\) Quotient \(=5m-3.\) Remainder \(=10m+8\) (degree \(1<3\)).

1. \((y^2+10y+24)\div(y+4)\)

Ans: Factor: \(y^2+10y+24=(y+4)(y+6)\). Quotient \(=y+6,\) remainder \(=0.\)

2. \((p^2+7p-5)\div(p+3)\)

Ans (long division): \(p^2\div p = p\Rightarrow p(p+3)=p^2+3p.\) Subtract → \(4p-5.\) \(4p\div p=4\Rightarrow4(p+3)=4p+12.\) Subtract → remainder \(-17.\) Quotient \(=p+4,\) remainder \(=-17.\)

3. \((3x+2x^2+4x^3)\div(x-4)\)

Ans: Arrange: \(4x^3+2x^2+3x.\) Step1: \(4x^3\div x=4x^2\Rightarrow\) subtract \(4x^3-16x^2\) → \(18x^2+3x\). Step2: \(18x^2\div x=18x\Rightarrow\) subtract \(18x^2-72x\) → \(75x\). Step3: \(75x\div x=75\Rightarrow\) subtract \(75x-300\) → remainder \(300\). Quotient \(=4x^2+18x+75,\) remainder \(=300.\)

4. \((2m^3+m^2+m+9)\div(2m-1)\)

Ans: Long division: \(2m^3\div2m=m^2\Rightarrow m^2(2m-1)=2m^3-m^2.\) Subtract → \(2m^2+m.\) \(2m^2\div2m=m\Rightarrow m(2m-1)=2m^2-m.\) Subtract → \(2m+9.\) \(2m\div2m=1\Rightarrow1(2m-1)=2m-1.\) Subtract → remainder \(10.\) Quotient \(=m^2+m+1,\) remainder \(=10.\)

5. \((3x-3x^2-12+x^4+x^3)\div(2+x^2)\)

Ans: Arrange: \(x^4+x^3-3x^2+3x-12\). \(x^4\div x^2=x^2\Rightarrow x^2(x^2+2)=x^4+2x^2.\) Subtract → \(x^3-5x^2+3x-12\). \(x^3\div x^2=x\Rightarrow x(x^2+2)=x^3+2x.\) Subtract → \(-5x^2+x-12\). \(-5x^2\div x^2=-5\Rightarrow -5(x^2+2)=-5x^2-10.\) Subtract → remainder \(x-2.\) Quotient \(=x^2+x-5,\) remainder \(=x-2.\)

6. \((a^4-a^3+a^2-a+1)\div(a^3-2)\)

Ans: \(a^4\div a^3=a\Rightarrow a(a^3-2)=a^4-2a.\) Subtract → \(-a^3+a^2+a+1\). \(-a^3\div a^3=-1\Rightarrow -1(a^3-2)=-a^3+2.\) Subtract → remainder \(a^2+a-1.\) Quotient \(=a-1,\) remainder \(=a^2+a-1.\)

7. \((4x^4-5x^3-7x+1)\div(4x-1)\)

Ans (fractional coefficients allowed): Long division yields quotient \(=x^3-x^2-\tfrac{1}{4}x-\tfrac{29}{16}\), remainder \(=-\tfrac{13}{16}.\)