JEE Mathematics Polynomials Notes Set 02

Important Notes and Formulas

Multiplying Polynomials

Monomials are multiplied by multiplying the coefficients and adding the powers of each variable, for example \( 6a^2b^4 \times 5a^4b^6 = 30a^6b^{10} \).
To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial. To multiply two polynomials, multiply every term of one by every term of the other and collect like terms.
Arrange both polynomials in descending powers of the variable and write the partial products one below the other, with like powers in the same column. This keeps the work tidy and makes errors easy to spot.

Products by Inspection

For linear factors, the middle term is the sum of the two constants and the last term is their product:
\( (x+a)(x+b) = x^2 + (a+b)x + ab \)
\( (x+a)(x+b)(x+c) = x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc \)
To find one particular coefficient of a product, list only the pairs of terms whose powers add up to the required power, multiply each pair and add. The full product is not needed.

Factorising Quadratic Expressions

To factorise \( x^2 + px + q \), find two numbers whose sum is \( p \) and whose product is \( q \). If these are \( a \) and \( b \), the factors are \( (x+a)(x+b) \).
Perfect squares: \( a^2 \pm 2ab + b^2 = (a \pm b)^2 \). Difference of squares: \( a^2 - b^2 = (a+b)(a-b) \).

Division of Polynomials

To divide one polynomial by another, arrange both in descending powers. Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient, multiply the divisor by it, subtract, and repeat with the new dividend until the remainder has a lower degree than the divisor.
The result is summarised by \( \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \). If the remainder is zero, the divisor is a factor of the dividend.

Remainder and Factor Theorems

When a polynomial \( f(x) \) is divided by \( x - c \), the remainder is \( f(c) \). For division by \( px - q \), the remainder is \( f(q/p) \).
So \( x - c \) is a factor of \( f(x) \) exactly when \( f(c) = 0 \). This is often the quickest way to find an unknown constant in a polynomial that is known to be divisible by a given factor.

Identities

An identity is an equation that holds for all values of the variables. To find unknown constants in an identity, either substitute convenient values of the variable, or compare the coefficients of like powers on both sides.
Useful identities: \( (a^2+b^2)(x^2+y^2) = (ax+by)^2 + (bx-ay)^2 \); \( (a+b)^2 + (a-b)^2 = 2(a^2+b^2) \); \( (a+b)^2 - (a-b)^2 = 4ab \); \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \).
Partial fractions: an expression such as \( \dfrac{1}{(x+1)(x+2)} \) can be written as \( \dfrac{A}{x+1} + \dfrac{B}{x+2} \). Clear the denominators and substitute the roots of the denominators to find \( A \) and \( B \).

Solved Examples

Example 1. Multiply \( 3x^2 - 7y^3 \) by \( -2x^2y \).
Answer:
Multiply each term: \( (3x^2)(-2x^2y) = -6x^4y \) and \( (-7y^3)(-2x^2y) = 14x^2y^4 \).
The product is \( -6x^4y + 14x^2y^4 \).

Example 2. Multiply \( 3x^2 + 5x - 1 \) by \( 4x - 3 \).
Answer:
Multiply by \( 4x \): \( 12x^3 + 20x^2 - 4x \).
Multiply by \( -3 \): \( -9x^2 - 15x + 3 \).
Adding like terms, the product is \( 12x^3 + 11x^2 - 19x + 3 \).

Example 3. Find the product of \( x^4 - 4x^3 + 6x^2 - 4x + 1 \) and \( x^2 - 2x + 1 \).
Answer:
Multiply by \( x^2 \), then by \( -2x \), then by \( 1 \) and add the rows with like powers aligned.
The product is \( x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 \).
This is \( (x-1)^4 (x-1)^2 = (x-1)^6 \), which provides a check.

Example 4. Find the coefficient of \( x \) in the product \( (x+2)(x-3)(x+4) \).
Answer:
Use the coefficient of \( x \) in \( (x+a)(x+b)(x+c) \), which is \( ab+bc+ca \), with \( a=2,\ b=-3,\ c=4 \).
\( ab+bc+ca = -6 - 12 + 8 = -10 \).

Example 5. Find the coefficient of \( x^2 \) in \( (2x^2 - 3ax + a^2)(-x^2 + 3ax - 2a^2) \) without full multiplication.
Answer:
The terms giving \( x^2 \) are \( (2x^2)(-2a^2) \), \( (-3ax)(3ax) \) and \( (a^2)(-x^2) \).
Their sum is \( -4a^2 - 9a^2 - a^2 = -14a^2 \).

Example 6. Find the coefficient of \( x^6 \) in \( (1 + 2x + 3x^2 + 4x^3 + 5x^4)(2 + 4x + 6x^2 + 8x^3) \).
Answer:
Pairs of terms with powers adding to 6: \( (4x^3)(8x^3) \) and \( (5x^4)(6x^2) \).
Coefficient \( = 32 + 30 = 62 \).

Example 7. The coefficient of \( x^2 \) in \( (3x^2 + ax + 2)(4x^2 + 2x + a) \) is \( -17 \). Find \( a \).
Answer:
The \( x^2 \) terms come from \( (3x^2)(a) \), \( (ax)(2x) \) and \( (2)(4x^2) \).
The coefficient is \( 3a + 2a + 8 = 5a + 8 \).
\( 5a + 8 = -17 \) gives \( a = -5 \).

Example 8. Factorise \( x^2 + 7x + 10 \), \( x^2 - 6x + 8 \) and \( a^2 - 3ab - 10b^2 \).
Answer:
\( x^2 + 7x + 10 \): numbers with sum 7 and product 10 are 5 and 2, so \( (x+5)(x+2) \).
\( x^2 - 6x + 8 \): numbers with sum \( -6 \) and product 8 are \( -4 \) and \( -2 \), so \( (x-4)(x-2) \).
\( a^2 - 3ab - 10b^2 \): numbers with sum \( -3 \) and product \( -10 \) are \( -5 \) and \( 2 \), so \( (a-5b)(a+2b) \).

Example 9. Factorise \( (x^2-9)(x^2+4x+4) - (x^2-6x+9)(x^2-4) \).
Answer:
Write each factor: \( (x-3)(x+3)(x+2)^2 - (x-3)^2(x-2)(x+2) \).
Take out the common factor \( (x-3)(x+2) \): \( (x-3)(x+2)\left[(x+3)(x+2) - (x-3)(x-2)\right] \).
The bracket is \( (x^2+5x+6) - (x^2-5x+6) = 10x \).
So the expression equals \( 10x(x-3)(x+2) \).

Example 10. Divide \( 24x^2 - 65x + 21 \) by \( 8x - 3 \).
Answer:
\( 24x^2 \div 8x = 3x \); \( 3x(8x-3) = 24x^2 - 9x \); subtracting leaves \( -56x + 21 \).
\( -56x \div 8x = -7 \); \( -7(8x-3) = -56x + 21 \); subtracting leaves 0.
Quotient \( 3x - 7 \), remainder 0, so \( 8x - 3 \) is a factor.

Example 11. Divide \( 30x^4 + 11x^3 - 82x^2 - 5x + 3 \) by \( 3x^2 + 2x - 4 \).
Answer:
\( 30x^4 \div 3x^2 = 10x^2 \); subtract \( 30x^4 + 20x^3 - 40x^2 \) to leave \( -9x^3 - 42x^2 - 5x \).
\( -9x^3 \div 3x^2 = -3x \); subtract \( -9x^3 - 6x^2 + 12x \) to leave \( -36x^2 - 17x + 3 \).
\( -36x^2 \div 3x^2 = -12 \); subtract \( -36x^2 - 24x + 48 \) to leave \( 7x - 45 \).
Quotient \( 10x^2 - 3x - 12 \), remainder \( 7x - 45 \).

Example 12. If \( 2x^3 - 9x^2 + ax - 6 \) is exactly divisible by \( 2x - 3 \), find \( a \).
Answer:
By the factor theorem the polynomial vanishes at \( x = \dfrac32 \).
\( 2\cdot\dfrac{27}{8} - 9\cdot\dfrac94 + \dfrac{3a}{2} - 6 = 0 \), that is \( \dfrac{27}{4} - \dfrac{81}{4} + \dfrac{3a}{2} - 6 = 0 \).
So \( \dfrac{3a}{2} = 19.5 \) and \( a = 13 \). The quotient is then \( x^2 - 3x + 2 \).

Example 13. Find \( A \) and \( B \) if \( \dfrac{1}{(x+1)(x+2)} = \dfrac{A}{x+1} + \dfrac{B}{x+2} \) for all \( x \).
Answer:
Clear denominators: \( 1 = A(x+2) + B(x+1) \).
Put \( x = -1 \): \( 1 = A \), so \( A = 1 \). Put \( x = -2 \): \( 1 = -B \), so \( B = -1 \).

Example 14. Find \( a \) and \( b \) so that \( 4x + 7 = a(x-1) + b(x-2) \) is an identity.
Answer:
Put \( x = 1 \): \( 11 = -b \), so \( b = -11 \).
Put \( x = 2 \): \( 15 = a \), so \( a = 15 \).
Check: \( 15(x-1) - 11(x-2) = 4x + 7 \).

Example 15. Find \( N \) such that \( a(b^2 - c^2) + b(c^2 - a^2) + c(a^2 - b^2) = N(b-c)(c-a)(a-b) \) for all \( a,b,c \).
Answer:
Substitute \( a=1,\ b=2,\ c=3 \). Left side: \( 1(4-9) + 2(9-1) + 3(1-4) = -5 + 16 - 9 = 2 \).
Right side: \( N(2-3)(3-1)(1-2) = 2N \).
So \( 2N = 2 \) and \( N = 1 \).

Example 16. Prove that \( (a^2+b^2)(x^2+y^2) = (ax+by)^2 + (bx-ay)^2 \).
Answer:
Expand the right side: \( (a^2x^2 + 2abxy + b^2y^2) + (b^2x^2 - 2abxy + a^2y^2) \).
The terms \( 2abxy \) cancel, leaving \( a^2x^2 + b^2y^2 + b^2x^2 + a^2y^2 \).
This equals \( (a^2+b^2)(x^2+y^2) \).