Maharashtra Board Class 8 Maths Chapter 10 Division of Polynomials Set 10.2 Solutions

Question 1. (i) (y² + 10y + 24) ÷ (y + 4)
Answer:
Quotient = \( y + 6 \)
Remainder = \( 0 \)
Explanation:
(i) \( (y+4) \times y = y^2 + 4y \)
(ii) \( (y+4) \times 6 = 6y + 24 \)
In simple words: This polynomial division involves dividing \( y^2 + 10y + 24 \) by \( y + 4 \). The process yields a quotient of \( y + 6 \) and a remainder of \( 0 \), meaning \( y + 4 \) is a factor of the dividend.

🎯 Exam Tip: Remember to align terms correctly by their powers during polynomial long division to avoid errors. A remainder of zero indicates a perfect division.

Question 1. (ii) (p² + 7p - 5) ÷ (p + 3)
Answer:
Quotient = \( p + 4 \)
Remainder = \( -17 \)
Explanation:
(i) \( (p+3) \times p = p^2 + 3p \)
(ii) \( (p+3) \times 4 = 4p + 12 \)
In simple words: When dividing the polynomial \( p^2 + 7p - 5 \) by \( p + 3 \), the result is a quotient of \( p + 4 \) and a remainder of \( -17 \), indicating that the division is not exact.

🎯 Exam Tip: Pay close attention to signs when subtracting terms in polynomial division. A non-zero remainder means the divisor is not a factor of the dividend.

Question 1. (iii) (3x + 2x² + 4x³) ÷ (x - 4)
Answer:
Write the dividend in descending order of their indices.
\( 3x + 2x^2 + 4x^3 = 4x^3 + 2x^2 + 3x \)
Quotient = \( 4x^2 + 18x + 75 \)
Remainder = \( 300 \)
Explanation:
(i) \( (x-4) \times 4x^2 = 4x^3 - 16x^2 \)
(ii) \( (x-4) \times 18x = 18x^2 - 72x \)
(iii) \( (x-4) \times 75 = 75x - 300 \)
In simple words: After arranging the dividend in descending powers of x as \( 4x^3 + 2x^2 + 3x \), the division by \( x - 4 \) results in a quotient of \( 4x^2 + 18x + 75 \) and a remainder of \( 300 \).

🎯 Exam Tip: Always reorder polynomials by descending powers of the variable before starting division to ensure correct alignment and calculation.

Question 1. (iv) (2m³ + m² + m + 9) ÷ (2m - 1)
Answer:
Quotient = \( m^2 + m + 1 \)
Remainder = \( 10 \)
Explanation:
(i) \( (2m-1) \times m^2 = 2m^3 - m^2 \)
(ii) \( (2m-1) \times m = 2m^2 - m \)
(iii) \( (2m-1) \times 1 = 2m - 1 \)
In simple words: Dividing the polynomial \( 2m^3 + m^2 + m + 9 \) by \( 2m - 1 \) results in a quotient of \( m^2 + m + 1 \) and a remainder of \( 10 \).

🎯 Exam Tip: When dividing by a binomial like \( (2m-1) \), ensure you correctly determine the term needed in the quotient at each step to eliminate the leading term of the remaining dividend.

Question 1. (v) (3x - 3x² - 12 + x⁴ + x³) ÷ (2 + x²)
Answer:
Write the dividend in descending order of their indices.
\( (x^4 + x^3 - 3x^2 + 3x - 12) \div (x^2 + 2) \)
Quotient = \( x^2 + x - 5 \)
Remainder = \( x - 2 \)
Explanation:
(i) \( (x^2+2) \times x^2 = x^4+2x^2 \)
(ii) \( (x^2+2) \times x = x^3 + 2x \)
(iii) \( (x^2+2) \times -5 = -5x^2-10 \)
In simple words: After arranging the dividend as \( x^4 + x^3 - 3x^2 + 3x - 12 \), the division by \( x^2 + 2 \) yields a quotient of \( x^2 + x - 5 \) and a remainder of \( x - 2 \).

🎯 Exam Tip: Always ensure both the dividend and divisor are in descending order of powers. For missing terms, use a coefficient of zero (e.g., \( 0x^2 \)) to maintain proper column alignment during division.

Question 1. (vi) (a⁴ - a³ + a² - a + 1) ÷ (a³ - 2)
Answer:
Quotient = \( a - 1 \)
Remainder = \( a^2 + a - 1 \)
Explanation:
(i) \( (a^3-2) \times a = a^4-2a \)
(ii) \( (a^3-2) \times -1 = -a^3+2 \)
In simple words: When performing the polynomial division of \( a^4 - a^3 + a^2 - a + 1 \) by \( a^3 - 2 \), the computed quotient is \( a - 1 \) and the remainder obtained is \( a^2 + a - 1 \).

🎯 Exam Tip: Be cautious with divisions involving different powers in the divisor and dividend. Missing terms in the dividend should be represented with zero coefficients for clarity.

Question 1. (vii) (4x⁴ - 5x³ - 7x + 1) ÷ (4x - 1)
Answer:
Write the dividend in descending order of their indices.
\( (4x^4 - 5x^3 - 7x + 1) = (4x^4 - 5x^3 + 0x^2 - 7x + 1) \)
Quotient = \( x^3 - x^2 - \frac{x}{4} - \frac{29}{16} \)
Remainder = \( -\frac{13}{16} \)
Explanation:
(i) \( (a^3-2) \times a = a^4-2a \)
(ii) \( (a^3-2) \times -1 = -a^3+2 \)
In simple words: To divide \( 4x^4 - 5x^3 - 7x + 1 \) by \( 4x - 1 \), first rearrange the dividend with a \( 0x^2 \) term. The division results in a quotient of \( x^3 - x^2 - \frac{x}{4} - \frac{29}{16} \) and a remainder of \( -\frac{13}{16} \).

🎯 Exam Tip: When dealing with fractional quotients or remainders, ensure all calculations for coefficients are precise. Remember to add zero coefficients for missing terms in the dividend for accurate long division.