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UNIT 1
: PRECALCULUS PREP
LESSON 4: OPERATIONS WITH POLYNOMIALS
Addition and Subtraction:
Example 1: “+” sign preceding
brackets – simply drop the brackets and collect like terms
a) (3x2 – 2x + 5) + (5x2 – 3x – 6) = 3x2 – 2x + 5 + 5x2 – 3x – 6 ** drop brackets
= 3x2 + 5x2 –2x – 3x +5 – 6 ** Collect like terms
= 8x2 – 5x – 1
Example 2: “-” sign preceding
brackets – multiply each term in the bracket by “-1” and collect like
terms
(5x2 – 3x + 6) – (2x2 – 7x + 8) = (5x2 – 3x + 6) – 1(2x2 – 7x + 8) ** multiply 2nd bracket by “-1”
= 5x2 – 3x + 6 – 2x2 + 7x - 8
= 5x2 – 2x2 – 3x + 7x + 6 – 8 ** collect like terms
= 3x2 + 4x - 2
Example 3:
a) (7a2 – 3ab – 5b2) + ( 4a2 – 8ab – 2b2) = 7a2 – 3ab – 5b2 + 4a2 – 8ab – 2b2
= 7a2 + 4a2 – 3ab – 8ab – 5b2 – 2b2
= 11a2 – 11ab – 7b2
b) (8x + 3y – 5xy) – (4x – 7y – 6xy) = (8x + 3y – 5xy) – 1(4x – 7y – 6xy) ** multiply 2nd bracket by “-1”
= 8x + 3y – 5xy – 4x + 7y + 6xy
= 8x – 4x + 3y + 7y – 5xy + 6xy ** collect like terms
= 4x + 10y + xy
Multiplying with Polynomials (Expanding):
Example 1: Monomial x Polynomial
– multiply each term in bracket by the monomial
a) –3x(2x2 – 5x + 7) = -3x(2x2) –
3x(-5x) – 3x(7) ** multiply each term by “-3x”
= -6x3 + 15x2 – 21x
b) 2x(5x – 3) –5(2x
+ 7) = 10x2 – 6x – 10x – 35 ** multiply each term in 1st bracket by “2x” and 2nd
bracket by “-5”
= 10x2 – 16x – 35 ** collect
like terms
Example 2: Polynomial x Polynomial – multiply each term in 1st bracket by each term in 2nd bracket
a) (3x + 5)(2x – 7) = 3x(2x – 7) + 5(2x – 7) ** multiply each term in 1st bracket by each term in 2nd bracket
= 6x2 – 21x + 10x –35 ** expand as in previous example
b) (2x + 3)(3x2 – 5x – 2) = 2x(3x2 –
5x – 2) + 3(3x2 – 5x – 2) ** multiply each term in 1st bracket by each term in 2nd bracket
= 6x3 –10x2 – 4x + 9x2
–15x – 6 ** expand
= 6x3 – x2 –19x – 6 ** collect like terms
c) 2(5x – 3)2 – 3(2x – 1)(3x + 2) = 2(5x – 3)(5x – 3) – 3(2x – 1)(3x + 2)
= 2(25x2 – 15x – 15x + 9) – 3(6x2 + 4x – 3x – 2)
= 50x2 – 30x – 30x + 18 – 18x2 – 12x + 9x + 6
= 32x2 – 63x + 24