JEE Mathematics Addition and Subtraction of Polynomials Notes Set 01

Like Terms and Addition

Terms whose variable parts are identical and differ only in the numerical coefficient are like terms: \( 2a \) and \( 5a \); \( 5a^2b^2 \) and \( -9a^2b^2 \). Unlike pairs are, for instance, \( (2a, 5b) \), \( (a^3, 2a^2b) \), \( (9xy, 5x^2y) \). Only like terms can be added.
The algebraic sum of like terms is found by adding the coefficients: \( -6x + 2x + 7x + 11x = (-6 + 2 + 7 + 11)x = 14x \), and \( 5a - 2a + 6a - a + 4a = 12a \).
There are two layouts. (1) Group the like terms in brackets and add. (2) Write the polynomials one under another with like terms in the same column, in descending powers, and add column by column. Check any result by substituting a convenient value such as x = 1.

Example 1. Simplify \( (3x - 2y + 7z) + (2x + 4y - z) + (x - 3y - 2z) \).
Answer:
\( x: 3 + 2 + 1 = 6 \); \( y: -2 + 4 - 3 = -1 \); \( z: 7 - 1 - 2 = 4 \).
Sum \( = \)\( 6x - y + 4z \).

Example 2. Add \( 2 - 4x + x^2 - 3x^3 \), \( 1 - 2x^2 + x^3 \) and \( 3x + x^2 - 5x^3 \).
Answer:
\( x^3: -3 + 1 - 5 = -7 \); \( x^2: 1 - 2 + 1 = 0 \); \( x: -4 + 3 = -1 \); constants: \( 2 + 1 = 3 \).
Sum \( = \)\( -7x^3 - x + 3 \).
Check at x = 1: the three values are -4, 0 and -1, total -5; and \( -7 - 1 + 3 = -5 \).

Example 3. Find the sum of \( x^2 - 3xy + y^2 + x + y - 1 \), \( 2x^2 + 4xy - 3y^2 - 2x - 2y + 3 \), \( 3x^2 - 5xy - 4y^2 + 3x + 4y - 2 \), \( 6x^2 + 10xy + 5y^2 + x + y \).
Answer:
\( x^2: 12 \); \( xy: -3 + 4 - 5 + 10 = 6 \); \( y^2: 1 - 3 - 4 + 5 = -1 \); \( x: 1 - 2 + 3 + 1 = 3 \); \( y: 1 - 2 + 4 + 1 = 4 \); constants: \( -1 + 3 - 2 = 0 \).
Sum \( = \)\( 12x^2 + 6xy - y^2 + 3x + 4y \).
Check at x = 1, y = 2: the four values are 1, -5, -14, 49, total 31; the sum gives \( 12 + 12 - 4 + 3 + 8 = 31 \).

Example 4. Add \( 3xy^2 - 2y^3 + x^3 \), \( 5x^2y - xy^2 - 3x^3 \), \( 8x^3 + 5y^3 \), \( 9x^2y - 2x^3 - xy^2 \).
Answer:
\( x^3: 1 - 3 + 8 - 2 = 4 \); \( x^2y: 5 + 9 = 14 \); \( xy^2: 3 - 1 - 1 = 1 \); \( y^3: -2 + 5 = 3 \).
Sum \( = \)\( 4x^3 + 14x^2y + xy^2 + 3y^3 \).

Subtraction

To subtract one polynomial from another, change the sign of every term of the polynomial being subtracted and add. In the column layout, write the polynomial to be subtracted below, with signs changed.

Example 5. Subtract \( 5a - 3b - 2c \) from \( 6a - 5b + 7c \).
Answer:
\( 6a - 5b + 7c - 5a + 3b + 2c = \)\( a - 2b + 9c \).

Example 6. Subtract \( 1 - 3x + 3x^2 - x^3 \) from \( x^3 - 3x^2 + 3x - 1 \).
Answer:
\( x^3 + x^3 = 2x^3 \); \( -3x^2 - 3x^2 = -6x^2 \); \( 3x + 3x = 6x \); \( -1 - 1 = -2 \).
Result: \( 2x^3 - 6x^2 + 6x - 2 \).

Example 7. If \( x = a^2 + 2ab + b^2 \), \( y = a^2 - 2ab + b^2 \), \( z = a^2 - b^2 \), find \( x + y - z \).
Answer:
\( x + y = 2a^2 + 2b^2 \). Subtract \( z \): \( 2a^2 + 2b^2 - a^2 + b^2 = \)\( a^2 + 3b^2 \).

Example 8. Find the polynomial which, added to \( x^2 + 3y^2 - 2z^2 + 5xy - 5yz + 3zx \), gives \( 3x^2 + 3y^2 - 2z^2 + 10xy + 5yz - zx \).
Answer:
Required polynomial = second minus first.
\( x^2: 3 - 1 = 2 \); \( y^2: 0 \); \( z^2: 0 \); \( xy: 10 - 5 = 5 \); \( yz: 5 + 5 = 10 \); \( zx: -1 - 3 = -4 \).
Answer: \( 2x^2 + 5xy + 10yz - 4zx \).

Example 9. Which polynomial must be subtracted from \( -4y^3 + 3xy^2 - x^2y - 5 \) so that the remainder is \( 16 - xy^2 + y^3 \)?
Answer:
Required polynomial = given minus remainder.
\( (-4y^3 + 3xy^2 - x^2y - 5) - (y^3 - xy^2 + 16) = -5y^3 + 4xy^2 - x^2y - 21 \).
Answer: \( -5y^3 + 4xy^2 - x^2y - 21 \).

Example 10. Add \( a - \frac12 b + c \), \( b - \frac12 c + a \), \( c - \frac12 a + b \) and subtract the result from 0.
Answer:
Sum: each of \( a, b, c \) has coefficient \( 1 + 1 - \frac12 = \frac32 \), so the sum is \( \frac32(a + b + c) \).
\( 0 - \frac32(a+b+c) = \)\( -\frac32 a - \frac32 b - \frac32 c \).

Example 11. What expression subtracted from 1 gives the sum of \( x - \frac12 y - z \), \( y - \frac12 z - x \), \( z - \frac12 x - y \)?
Answer:
Sum: coefficient of each of \( x, y, z \) is \( 1 - 1 - \frac12 = -\frac12 \), so the sum is \( -\frac12(x + y + z) \).
\( 1 - E = -\frac12(x+y+z) \), hence \( E = 1 + \frac12 x + \frac12 y + \frac12 z \).

Example 12. By how much does the sum of \( 2x^2 - 3xy + 4y^2 \) and \( 5y^2 - 3xy - x^2 \) exceed \( 5x^2 - y^2 \)?
Answer:
Sum \( = x^2 - 6xy + 9y^2 \).
Excess \( = x^2 - 6xy + 9y^2 - 5x^2 + y^2 = \)\( -4x^2 - 6xy + 10y^2 \).

Example 13. What quantity is less than \( 2x^2 - 3x + 5 \) by \( 3x^2 - 7x - 1 \)?
Answer:
Let the quantity be \( Q \). Then \( (2x^2 - 3x + 5) - Q = 3x^2 - 7x - 1 \).
\( Q = (2x^2 - 3x + 5) - (3x^2 - 7x - 1) = \)\( -x^2 + 4x + 6 \).

Example 14. The sum of \( 3x^4 + 5x^2 - 11x + 13 \) and \( ax^4 + bx^3 + cx^2 + dx + e \) is \( 7x^4 + 9x^3 - 2x^2 + 6x - 9 \). Find a, b, c, d, e.
Answer:
Compare coefficients: \( 3 + a = 7 \Rightarrow a = 4 \); \( b = 9 \); \( 5 + c = -2 \Rightarrow c = -7 \); \( -11 + d = 6 \Rightarrow d = 17 \); \( 13 + e = -9 \Rightarrow e = -22 \).
Answer: a = 4, b = 9, c = -7, d = 17, e = -22.

Example 15. A has x marbles, B has y times as many as A, and C has 5 fewer than B. Write the total in x and y, and find it for x = 10, y = 3.
Answer:
A: \( x \); B: \( xy \); C: \( xy - 5 \). Total \( = x + 2xy - 5 \).
For x = 10, y = 3: \( 10 + 60 - 5 = \)65 marbles.