[MODULE 5:
SOLVING QUADRATIC EQUATIONS BY FACTORING
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS]
Name: ___________________________________
New York City College of Technology
MAT 1175 PAL Workshops
Points: ______
1. Solving Quadratic Equations by Factoring
To solve a quadratic equation by factoring:
1) Put the equation in standard form: ax  bx  c  0 .
2) Factor completely.
3) Use the zero-product rule, set each factor containing the variable equal to zero and solve for x.
Note: Do not solve for the constant factor.
2
2
a) Solve for x: 2 x  18  0
2
b) Solve for x: 2 x  18 x  0
3
2
c) Solve for x: 2 x  9 x  5 x
d) Solve for x: x(5 x  2)  3
2. Simplifying Rational Expressions
A rational expression is an expression that can be written in the form
P
, where P and Q are polynomials and
Q
Q0
To simplifying rational expressions to lowest terms
1. Factor the polynomial in the numerator and denominator.
2. Apply the fundamental principle of rational expressions to divide out factors common to both the
numerator and denominator.
PR P
where R  0

QR Q
1
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
[Fundamental Principle of Rational Expressions]
[MODULE 5:
SOLVING QUADRATIC EQUATIONS BY FACTORING
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS]
a) Simplify:
p2
p2
b) Simplify:
p2
2 p
c) Simplify:
p2
p2
d) Simplify:
p2
2 p
New York City College of Technology
MAT 1175 PAL Workshops
2y2  8y
e) Simplify:
3 y 2  12 y
3. Multiplying Rational Expressions
Recall in fractions:
12 15 3 3 9

  
25 16 5 4 20
To Multiply Rational Expressions:
1. Factor all the polynomials in the numerators and denominators.
2. Multiply the numerators and multiply the denominators.
3. Simplify the product by applying the fundamental property and dividing the numerator and denominator
by their common factors.
a) Multiply:
c) Multiply:
2
12 24

18 6
x 2  64 x  3

3  x x 2  8x
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
b) Multiply:
15a 4 21b 5

14a 5 b 25ab
[MODULE 5:
d) Multiply:
SOLVING QUADRATIC EQUATIONS BY FACTORING
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS]
New York City College of Technology
MAT 1175 PAL Workshops
12a 2  18a 4a 2  8a  3

4a 2  4a  1
4a 2  9
4. Dividing Rational Expressions
Recall in fractions:
12 16 12 15 3 3 9



  
25 15 25 16 5 4 20
To Divide Rational Expressions:
1. Change the division sign to a multiplication sign.
2. Take the reciprocal of the second rational expression (flip the rational expression)
3. Follow the steps used in multiplying rational expressions.
a) Divide:
24 p 3 q 4 36 p 3 q7

12 p 4 q
6 pq 6
b) Divide:
m6n3
 4m 5 n 4
25m 4 n
3
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
[MODULE 5:
c) Divide:
x 2  36 3x  18

4 x  x 2 x 2  16
e) Multiply and divide:
4
SOLVING QUADRATIC EQUATIONS BY FACTORING
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS]
3x 2  5 x  2 x 2  4 x  5 5 x 2  9 x  2


x 2  x  2 12 x 2  7 x  1 8 x 2  2 x  1
Prepared by Profs. Ghezzi, Han, and Liou-Mark
Supported by BMI, NSF STEP #0622493, and Perkins
New York City College of Technology
MAT 1175 PAL Workshops