Maharashtra Board Class 9 Maths Chapter 3 Set 3.3 Algebra Standard Part 1 Polynomials Solutions

Question 1. Divide each of the following polynomials by synthetic division method and also by linear division method. Write the quotient and the remainder.
i. \( (2m^2 - 3m + 10) \div (m - 5) \)
ii. \( (x^4 + 2x^3 + 3x^2 + 4x + 5) \div (x + 2) \)
iii. \( (y^3 - 216) \div (y - 6) \)
iv. \( (2x^4 + 3x^3 + 4x - 2x^2) \div (x + 3) \)
Answer:
i. \( (2m^2 - 3m + 10) \div (m - 5) \)
Synthetic Division Method:
Write the dividend polynomial in coefficient form: \( 2m^2 - 3m + 10 \)
Coefficient form = \( (2, -3, 10) \)
Divisor = \( m - 5 \); take the opposite of \( -5 \), which is \( 5 \).
Performing synthetic division:
We write the divisor \( 5 \) on the left, and the coefficients \( 2, -3, 10 \) in a row.
Bring down the first coefficient \( 2 \). Multiply \( 2 \times 5 = 10 \). Add to \( -3 \) to get \( 7 \).
Multiply \( 7 \times 5 = 35 \). Add to \( 10 \) to get \( 45 \).
The last number \( 45 \) is the remainder.
The quotient coefficients are \( (2, 7) \).
Quotient = \( 2m + 7 \)
Remainder = \( 45 \)

Linear Division Method:
\( 2m^2 - 3m + 10 = 2m(m - 5) + 10m - 3m + 10 \)
\( = 2m(m - 5) + 7m + 10 \)
\( = 2m(m - 5) + 7(m - 5) + 35 + 10 \)
\( = (2m + 7)(m - 5) + 45 \)
Quotient = \( 2m + 7 \)
Remainder = \( 45 \)

ii. \( (x^4 + 2x^3 + 3x^2 + 4x + 5) \div (x + 2) \)
Synthetic Division Method:
Dividend = \( x^4 + 2x^3 + 3x^2 + 4x + 5 \)
Coefficient form = \( (1, 2, 3, 4, 5) \)
Divisor = \( x + 2 \); take the opposite of \( 2 \), which is \( -2 \).
Performing synthetic division:
Bring down \( 1 \). Multiply \( 1 \times (-2) = -2 \). Add to \( 2 \) to get \( 0 \).
Multiply \( 0 \times (-2) = 0 \). Add to \( 3 \) to get \( 3 \).
Multiply \( 3 \times (-2) = -6 \). Add to \( 4 \) to get \( -2 \).
Multiply \( -2 \times (-2) = 4 \). Add to \( 5 \) to get \( 9 \).
The quotient coefficients are \( (1, 0, 3, -2) \).
Quotient = \( x^3 + 3x - 2 \)
Remainder = \( 9 \)

Linear Division Method:
\( x^4 + 2x^3 + 3x^2 + 4x + 5 = x^3(x + 2) - 2x^3 + 2x^3 + 3x^2 + 4x + 5 \)
\( = x^3(x + 2) + 3x^2 + 4x + 5 \)
\( = x^3(x + 2) + 3x(x + 2) - 6x + 4x + 5 \)
\( = x^3(x + 2) + 3x(x + 2) - 2x + 5 \)
\( = x^3(x + 2) + 3x(x + 2) - 2(x + 2) + 4 + 5 \)
\( = (x^3 + 3x - 2)(x + 2) + 9 \)
Quotient = \( x^3 + 3x - 2 \)
Remainder = \( 9 \)

iii. \( (y^3 - 216) \div (y - 6) \)
Synthetic Division Method:
Dividend in index form = \( y^3 + 0y^2 + 0y - 216 \)
Coefficient form = \( (1, 0, 0, -216) \)
Divisor = \( y - 6 \); take the opposite of \( -6 \), which is \( 6 \).
Performing synthetic division:
Bring down \( 1 \). Multiply \( 1 \times 6 = 6 \). Add to \( 0 \) to get \( 6 \).
Multiply \( 6 \times 6 = 36 \). Add to \( 0 \) to get \( 36 \).
Multiply \( 36 \times 6 = 216 \). Add to \( -216 \) to get \( 0 \).
The quotient coefficients are \( (1, 6, 36) \).
Quotient = \( y^2 + 6y + 36 \)
Remainder = \( 0 \)

Linear Division Method:
\( y^3 - 216 = y^2(y - 6) + 6y^2 - 216 \)
\( = y^2(y - 6) + 6y(y - 6) + 36y - 216 \)
\( = y^2(y - 6) + 6y(y - 6) + 36(y - 6) + 216 - 216 \)
\( = (y^2 + 6y + 36)(y - 6) \)
Quotient = \( y^2 + 6y + 36 \)
Remainder = \( 0 \)

iv. \( (2x^4 + 3x^3 + 4x - 2x^2) \div (x + 3) \)
Synthetic Division Method:
Dividend in standard form = \( 2x^4 + 3x^3 - 2x^2 + 4x + 0 \)
Coefficient form = \( (2, 3, -2, 4, 0) \)
Divisor = \( x + 3 \); take the opposite of \( 3 \), which is \( -3 \).
Performing synthetic division:
Bring down \( 2 \). Multiply \( 2 \times (-3) = -6 \). Add to \( 3 \) to get \( -3 \).
Multiply \( -3 \times (-3) = 9 \). Add to \( -2 \) to get \( 7 \).
Multiply \( 7 \times (-3) = -21 \). Add to \( 4 \) to get \( -17 \).
Multiply \( -17 \times (-3) = 51 \). Add to \( 0 \) to get \( 51 \).
The quotient coefficients are \( (2, -3, 7, -17) \).
Quotient = \( 2x^3 - 3x^2 + 7x - 17 \)
Remainder = \( 51 \)

Linear Division Method:
\( 2x^4 + 3x^3 - 2x^2 + 4x = 2x^3(x + 3) - 6x^3 + 3x^3 - 2x^2 + 4x \)
\( = 2x^3(x + 3) - 3x^3 - 2x^2 + 4x \)
\( = 2x^3(x + 3) - 3x^2(x + 3) + 9x^2 - 2x^2 + 4x \)
\( = 2x^3(x + 3) - 3x^2(x + 3) + 7x^2 + 4x \)
\( = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 21x + 4x \)
\( = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 17x \)
\( = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 17(x + 3) + 51 \)
\( = (2x^3 - 3x^2 + 7x - 17)(x + 3) + 51 \)
Quotient = \( 2x^3 - 3x^2 + 7x - 17 \)
Remainder = \( 51 \)
In simple words: Synthetic division is a shortcut way to divide polynomials using only their coefficients, while linear division rearranges the terms step-by-step to factor out the divisor. Both methods always yield the exact same quotient and remainder.

🎯 Exam Tip: Always write the dividend polynomial in standard index form (including terms with zero coefficients) before starting synthetic division to avoid calculation errors.

Question. Divide the following polynomials using synthetic division method and linear division method:
(i) \( (2m^2 - 3m + 10) \div (m - 5) \)
(ii) \( (x^4 + 2x^3 + 3x^2 + 4x + 5) \div (x + 2) \)
(v) \( (x^4 - 3x^2 - 8) \div (x + 4) \)
(vi) \( (y^3 - 3y^2 + 5y - 1) \div (y - 1) \)
Answer:
(i) \( (2m^2 - 3m + 10) \div (m - 5) \)
Synthetic division:
Dividend = \( 2m^2 - 3m + 10 \)
\( \therefore \) Coefficient form of dividend = \( (2, -3, 10) \)
Divisor = \( m - 5 \)
\( \dots \) Opposite of \( -5 \) is \( 5 \).

Coefficient form of quotient = \( (2, 7) \)
\( \therefore \) Quotient = \( 2m + 7 \),
Remainder = \( 45 \)

Linear division method:
\( 2m^2 - 3m + 10 \)
To get the term \( 2m^2 \), multiply \( (m - 5) \) by \( 2m \) and add \( 10m \),
\( = 2m(m - 5) + 10m - 3m + 10 \)
\( = 2m(m - 5) + 7m + 10 \)
To get the term \( 7m \), multiply \( (m - 5) \) by \( 7 \) and add \( 35 \)
\( = 2m(m - 5) + 7(m - 5) + 35 + 10 \)
\( = (m - 5)(2m + 7) + 45 \)
\( \therefore \) Quotient = \( 2m + 7 \),
Remainder = \( 45 \)

(ii) \( (x^4 + 2x^3 + 3x^2 + 4x + 5) \div (x + 2) \)
Synthetic division:
Dividend = \( x^4 + 2x^3 + 3x^2 + 4x + 5 \)
\( \therefore \) Coefficient form of dividend = \( (1, 2, 3, 4, 5) \)
Divisor = \( x + 2 \)
\( \dots \) Opposite of \( +2 \) is \( -2 \).

Coefficient form of quotient = \( (1, 0, 3, -2) \)
In simple words: Synthetic division is a shortcut way of dividing polynomials using only their coefficients, while linear division involves rewriting the terms step-by-step to factor out the divisor.

🎯 Exam Tip: Always write the dividend in its complete index form (including any missing terms with coefficient 0) before writing the coefficient form to avoid calculation errors.

Question 1. Divide \( x^4 + 2x^3 + 3x^2 + 4x + 5 \) by \( x + 2 \) using the linear division method.
Answer:
Linear division method:
\( x^4 + 2x^3 + 3x^2 + 4x + 5 \)
To get the term \( x^4 \), multiply \( (x + 2) \) by \( x^3 \) and subtract \( 2x^3 \),
\( = x^3(x + 2) - 2x^3 + 2x^3 + 3x^2 + 4x + 5 \)
\( = x^3(x + 2) + 3x^2 + 4x + 5 \)
To get the term \( 3x^2 \), multiply \( (x + 2) \) by \( 3x \) and subtract \( 6x \),
\( = x^3(x + 2) + 3x(x + 2) - 6x + 4x + 5 \)
\( = x^3(x + 2) + 3x(x + 2) - 2x + 5 \)
To get the term \( -2x \), multiply \( (x + 2) \) by \( -2 \) and add \( 4 \),
\( = x^3(x + 2) + 3x(x + 2) - 2(x + 2) + 4 + 5 \)
\( = (x + 2)(x^3 + 3x - 2) + 9 \)

\( \therefore \) Quotient \( = x^3 + 3x - 2 \)
\( \text{Remainder} = 9 \)
In simple words: In the linear division method, we rewrite the polynomial step-by-step to show the divisor \( (x + 2) \) as a common factor, leaving us with the quotient and the remainder at the end.

🎯 Exam Tip: Always double-check your signs when subtracting terms in the linear division method to avoid calculation errors.

Question 2. Divide \( (y^3 - 216) \div (y - 6) \) by synthetic division and linear division methods.
Answer:
Synthetic division:
\( (y^3 - 216) \div (y - 6) \)
Dividend \( = y^3 - 216 \)

\( \therefore \) Index form \( = y^3 + 0y^2 + 0y - 216 \)

\( \dots \) Coefficient form of dividend \( = (1, 0, 0, -216) \)
Divisor \( = y - 6 \)

\( \therefore \) Opposite of \( -6 \) is \( 6 \).

Coefficient form of quotient \( = (1, 6, 36) \)

\( \therefore \) Quotient \( = y^2 + 6y + 36 \)
\( \text{Remainder} = 0 \)

Linear division method:
\( y^3 - 216 \)
To get the term \( y^3 \), multiply \( (y - 6) \) by \( y^2 \) and add \( 6y^2 \),
\( = y^2(y - 6) + 6y^2 - 216 \)
To get the term \( 6y^2 \), multiply \( (y - 6) \) by \( 6y \) and add \( 36y \),
\( = y^2(y - 6) + 6y(y - 6) + 36y - 216 \)
To get the term \( 36y \), multiply \( (y - 6) \) by \( 36 \) and add \( 216 \),
\( = y^2(y - 6) + 6y(y - 6) + 36(y - 6) + 216 - 216 \)
\( = (y - 6)(y^2 + 6y + 36) + 0 \)

\( \dots \) Quotient \( = y^2 + 6y + 36 \)
\( \text{Remainder} = 0 \)
In simple words: Synthetic division uses coefficients in a grid to find the quotient and remainder quickly, while linear division rearranges the terms step-by-step to factor out the divisor.

🎯 Exam Tip: Remember to write the dividend in index form with zero coefficients for any missing terms before starting synthetic division.

Question 1. Divide the polynomial \( (2x^4 + 3x^3 + 4x - 2x^2) \) by \( (x + 3) \) using synthetic division and linear division methods.
Answer:
iv. Synthetic division:
\( (2x^4 + 3x^3 + 4x - 2x^2) \div (x + 3) \)
Dividend = \( 2x^4 + 3x^3 + 4x - 2x^2 \)
\( \implies \) Index form = \( 2x^4 + 3x^3 - 2x^2 + 4x + 0 \)
\( \implies \) Coefficient form of the dividend = \( (2, 3, -2, 4, 0) \)
Divisor = \( x + 3 \)
\( \implies \) Opposite of \( +3 \) is \( -3 \)

Coefficient form of quotient = \( (2, -3, 7, -17) \)
\( \implies \) Quotient = \( 2x^3 - 3x^2 + 7x - 17 \)
Remainder = \( 51 \)

Linear division method:
\( 2x^4 + 3x^3 + 4x - 2x^2 = 2x^4 + 3x^3 - 2x^2 + 4x \)
To get the term \( 2x^4 \), multiply \( (x + 3) \) by \( 2x^3 \) and subtract \( 6x^3 \),
\( \implies = 2x^3(x + 3) - 6x^3 + 3x^3 - 2x^2 + 4x \)
\( \implies = 2x^3(x + 3) - 3x^3 - 2x^2 + 4x \)

To get the term \( -3x^3 \), multiply \( (x + 3) \) by \( -3x^2 \) and add \( 9x^2 \),
\( \implies = 2x^3(x + 3) - 3x^2(x + 3) + 9x^2 - 2x^2 + 4x \)
\( \implies = 2x^3(x + 3) - 3x^2(x + 3) + 7x^2 + 4x \)

To get the term \( 7x^2 \), multiply \( (x + 3) \) by \( 7x \) and subtract \( 21x \),
\( \implies = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 21x + 4x \)
\( \implies = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 17x \)

To get the term \( -17x \), multiply \( (x + 3) \) by \( -17 \) and add \( 51 \),
\( \implies = 2x^3(x + 3) - 3x^2(x + 3) + 7x(x + 3) - 17(x + 3) + 51 \)
\( \implies = (x + 3)(2x^3 - 3x^2 + 7x - 17) + 51 \)
In simple words: To divide a polynomial, we can use synthetic division by writing down only the coefficients and performing simple multiplication and addition. Alternatively, linear division breaks down the terms step-by-step to match the divisor, giving us the same quotient and remainder.

🎯 Exam Tip: Always write the dividend in its complete index form (including zero coefficients for missing terms) before starting synthetic division to avoid calculation errors.

Question 1. Divide the following polynomials using synthetic division method and linear division method.
(v) \( (x^4 - 3x^2 - 8) \div (x + 4) \)
(vi) \( (y^3 - 3y^2 + 5y - 1) \div (y - 1) \)
Answer:
(v) Synthetic division:
\( (x^4 - 3x^2 - 8) \div (x + 4) \)
Dividend = \( x^4 - 3x^2 - 8 \)
\( \therefore \) Index form = \( x^4 + 0x^3 - 3x^2 + 0x - 8 \)
\( \dots \) Coefficient form of the dividend = \( (1, 0, -3, 0, -8) \)
Divisor = \( x + 4 \)
\( \therefore \) Opposite of \( +4 \) is \( -4 \)

\( \therefore \) Coefficient form of quotient = \( (1, -4, 13, -52) \)
\( \therefore \) Quotient = \( x^3 - 4x^2 + 13x - 52 \),
Remainder = \( 200 \)

Linear division method:
\( x^4 - 3x^2 - 8 \)
To get the term \( x^4 \), multiply \( (x + 4) \) by \( x^3 \) and subtract \( 4x^3 \),
\( = x^3(x + 4) - 4x^3 - 3x^2 - 8 \)
\( = x^3(x + 4) - 4x^3 - 3x^2 - 8 \)
To get the term \( -4x^3 \), multiply \( (x + 4) \) by \( -4x^2 \) and add \( 16x^2 \),
\( = x^3(x + 4) - 4x^2(x + 4) + 16x^2 - 3x^2 - 8 \)
\( = x^3(x + 4) - 4x^2(x + 4) + 13x^2 - 8 \)
To get the term \( 13x^2 \), multiply \( (x + 4) \) by \( 13x \) and subtract \( 52x \),
\( = x^3(x + 4) - 4x^2(x + 4) + 13x(x + 4) - 52x - 8 \)
\( = x^3(x + 4) - 4x^2(x + 4) + 13x(x + 4) - 52x - 8 \)
To get the term \( -52x \), multiply \( (x + 4) \) by \( -52 \) and add \( 208 \),
\( = x^3(x + 4) - 4x^2(x + 4) + 13x(x + 4) - 52(x + 4) + 208 - 8 \)
\( = (x + 4)(x^3 - 4x^2 + 13x - 52) + 200 \)
\( \therefore \) Quotient = \( x^3 - 4x^2 + 13x - 52 \),
Remainder = \( 200 \)

(vi) Synthetic division:
\( (y^3 - 3y^2 + 5y - 1) \div (y - 1) \)
Dividend = \( y^3 - 3y^2 + 5y - 1 \)
Coefficient form of the dividend = \( (1, -3, 5, -1) \)
In simple words: Synthetic division is a shortcut method to divide polynomials using only their coefficients, while linear division involves rewriting terms step-by-step to factor out the divisor. Both methods yield the exact same quotient and remainder.

🎯 Exam Tip: Always write the dividend in its complete index form (including terms with zero coefficients) before starting synthetic division to avoid calculation errors.

Question. Divide \( y^3 - 3y^2 + 5y - 1 \) by \( y - 1 \) using the synthetic division method and the linear division method.
Answer:
Synthetic division method:
Divisor = \( y - 1 \)
\( \therefore \) Opposite of \( -1 \) is \( 1 \).

\( \therefore \) Coefficient form of quotient = \( (1, -2, 3) \)
\( \therefore \) Quotient = \( y^2 - 2y + 3 \)
Remainder = \( 2 \)

Linear division method:
\( y^3 - 3y^2 + 5y - 1 \)
To get the term \( y^3 \), multiply \( (y - 1) \) by \( y^2 \) and add \( y^2 \). This systematic step-by-step adjustment ensures that the value of the expression remains unchanged throughout the process.
\( = y^2(y - 1) + y^2 - 3y^2 + 5y - 1 \)
\( = y^2(y - 1) - 2y^2 + 5y - 1 \)
To get the term \( -2y^2 \), multiply \( (y - 1) \) by \( -2y \) and subtract \( 2y \),
\( = y^2(y - 1) - 2y(y - 1) - 2y + 5y - 1 \)
\( = y^2(y - 1) - 2y(y - 1) + 3y - 1 \)
To get the term \( 3y \), multiply \( (y - 1) \) by \( 3 \) and add \( 3 \),
\( = y^2(y - 1) - 2y(y - 1) + 3(y - 1) + 3 - 1 \)
\( = (y - 1)(y^2 - 2y + 3) + 2 \)
\( \therefore \) Quotient = \( y^2 - 2y + 3 \),
Remainder = \( 2 \).
In simple words: To divide a polynomial, we can use synthetic division by working only with the coefficients in a simple grid. Alternatively, the linear division method lets us break down and rewrite the polynomial step-by-step to factor out the divisor, giving us the exact same quotient and remainder.

🎯 Exam Tip: Always write down the final values of both the Quotient and the Remainder clearly at the end of your solution to secure full marks.