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UNIT
1 : ALGEBRA
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
Factoring Polynomials
Review of Basic Factoring methods:
1.
Common Factoring:
Factor 6x3
– 15x
Solution:
6x3 – 15x = 3x(2x2
– 5)
** Find the HIGHEST COMMON FACTOR for each term --- “3x”
** Divide “3x” into each term to get the second factor --- “2x2 – 5”
** Check by expanding
2.
Difference of Squares: Formula -- a2
– b2 = (a – b)(a + b)
Factor 49x2
– 64y2
Solution:
49x2 – 64y2 = (7x – 8y)(7x + 8y)
Factor x2
– 9y2
Solution:
x2 – 9y2
= (x – 3y)(x + 3y)
3.
Simple Trinomials: Form x2 + bx +
c [Coefficient of x2 is 1]
a) Factor x2
+ 5x + 6
Solution:
Recall x2 + 5x + 6 = (x + __ )(x + __ ) ** We
need two numbers that multiply to +6 and add to +5
** Check all pairs of factors of 6: {1,
6} adds to 7
{2, 3}
adds to 5
Hence x2
+ 5x + 6 = (x + 3 )(x + 2) ** Check by expanding
b) Factor x2
– 2x – 35
Solution:
Recall x2 - 2x - 35 = (x + __ )(x + __ ) ** We
need two numbers that multiply to -35 and add to -2
** Check all pairs of factors of 35: {1,
35} no
combination adds to “-2”
{5, 7} -7 + 5 = -2
Hence x2
– 2x - 35 = (x + 5 )(x – 7) ** Check by expanding
4.
Hard Trinomials: Form ax2 + bx +
c [Coefficient of x2 does not equal 1]
a) Factor
6m2 – 5m – 4
Solution:
We use the
method of decomposition (although there are other methods)
We
decompose the middle term “-5m” into two parts using the two clues:
Multiply to (6)(-4)
= -24 and
Add to “-5”
Strategy:
List all pairs of
factors of “-24” and find the pair that adds to “-5” ** {1, 24} cannot obtain “-5” with these two
factors; 1+24=25; 1-24= -23
** {2,
12} cannot obtain “-5” with these two factors
** {3,
8} -8 + 3 = -5 Choose these two factors
Hence 6m2 – 5m – 4 = 6m2 – 8m + 3m
–4 ** “-5m” broken into two parts “-8m + 3m”
= 2m(3m – 4) + 1(3m – 4) ** Group by twos and common factor
= (3m – 4)(2m + 1) ** check by expanding
b) Factor
2x2 – 9x + 4
Solution:
We
decompose the middle term “-9x” into two parts using the two clues:
Multiply to (2)(4)
= +8 and
Add
to “-9”
Strategy:
List all pairs of
factors of “+8” and find the pair that adds to “-9” ** {2, 4} cannot obtain “-9” with these two
factors; 2+4=6; 2-4= -2
** {1, 8}
-8 + -1 = -9 Choose these
two factors
Hence 2x2 – 9x + 4 = 2x2 –8x – 1x
+ 4 ** “-9x” broken into two parts “-8x – 1x”
= 2x(x – 4)
– 1(x – 4) ** Group by twos and common factor
= (x – 4)(2x – 1) ** check by expanding