Polynomials and Short Multiplication Formulas

Short Multiplication Formulas

Square of a sum and a difference:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b². In (a − b)² the last term is +b²
• a² − b² = (a − b)(a + b)
• From the two formulas above, the sum of squares: a² + b² = (a − b)² + 2ab, and also a² + b² = (a + b)² − 2ab.

Geometric meaning of the square: a large square with side a + b is divided into four parts.

Part	Area
square with side a	a²
two rectangles with sides a and b	ab and ab
square with side b	b²

Total: (a + b)² = a² + ab + ab + b².

Working with cubes:

Expression	Expansion
(a + b)³	a³ + 3a²b + 3ab² + b³
(a − b)³	a³ − 3a²b + 3ab² − b³
a³ + b³	(a + b)(a² − ab + b²)
a³ − b³	(a − b)(a² + ab + b²)

Working with Polynomials

• Order of factoring: first take the common factor out of the brackets, then apply a formula, and group terms if necessary.
• Bézout: the remainder when dividing by (x − a) is P(a).
• If P(a) = 0, then (x − a) is a factor of the polynomial, and a is its root.

1. Take the common factor out of the brackets
2. Look for a short multiplication formula
3. If the polynomial is long, group the terms in pairs
4. Multiply out and check that you get the original polynomial

Examples

• 101² − 99² = (101 − 99)(101 + 99) = 2 · 200 = 400. Also 99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
• x³ − 3x² − 4x + 12 = x²(x − 3) − 4(x − 3) = (x − 3)(x² − 4) = (x − 3)(x − 2)(x + 2). Check: expanding (x − 3)(x² − 4) gives x³ − 3x² − 4x + 12.
• When P(x) = x³ − 2x² + x − 5 is divided by (x − 2), the remainder is P(2) = 8 − 8 + 2 − 5 = −3, and the quotient is x² + 1. Check: (x − 2)(x² + 1) − 3 = x³ − 2x² + x − 5.

Be Careful

• When you open a bracket with a minus sign in front, the signs of all terms change: −(x − 3)² = −x² + 6x − 9.
• The remainder when dividing by (x + 2) is P(−2), not P(2).
• In the sum of cubes, the middle term of the second bracket is negative: a² − ab + b², not a² + ab + b².