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LESSON 5 : RATIONAL EXPRESSIONS PART 1
Rational expressions are quotients whose numerator and denominator are both polynomials [monomials, binomial, trinomial, etc…]
Examples:
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
Example 1:
Solutions: