BPM MATH0 – Week 1
Algebraic Expressions and Basic Equations
Week 1 – Algebraic Expressions and Basic Equations
Weekly Goals
ˆ Practice simplifying algebraic expressions involving powers, roots, and fractions.
ˆ Understand and apply the rules of exponents and radicals.
ˆ Manipulate and simplify rational expressions.
ˆ Solve linear and basic quadratic equations.
Solved Examples – With Detailed Steps
Example 1: Simplify the expression:
(a3
b4
c5
) ÷ (−a3
c2
)
Step-by-step:
ˆ Use the rule xm
xn = xm−n
.
ˆ Split: a3
−a3 = −1, b4
1
= b4
, c5
c2 = c3
ˆ Result: −b4
c3
−b4
c3
Example 2: Simplify:
1 + 1
x−1
1 − 1
x+1
ˆ Numerator: x−1
x−1
+ 1
x−1
= x
x−1
, Denominator: x+1
x+1
− 1
x+1
= x
x+1
ˆ Convert division to multiplication and cancel: x
x−1
÷ x
x+1
= x
x−1
· x+1
x
= x+1
x−1
x + 1
x − 1
Example 3: Solve:
2x2
− 12x + 36 = x2
+ 2x − 13
ˆ Rearranged: x2
− 14x + 49 = 0
ˆ Factor: (x − 7)2
= 0 ⇒ x = 7
x = 7
Example 4: Simplify:
x2
− 9
x2 + 5x + 6
·
x + 2
x − 3
ˆ Factor: (x−3)(x+3)
(x+2)(x+3)
· x+2
x−3
ˆ Cancel all terms
1
BPM MATH0 – Week 1
Algebraic Expressions and Basic Equations
Example 5: Simplify:
1
√
x + 1
−
√
x
x − 1
ˆ Rationalize first term: 1√
x+1
·
√
x−1
√
x−1
=
√
x−1
x−1
ˆ Subtract:
√
x−1−
√
x
x−1
= −1
x−1
−
1
x − 1
Example 6: Simplify:
3
a ·
√
b
3
√
a · b2
ˆ Convert radicals to powers:
a1/3
(b1/2
)1/3
(a1/2)1/3(b2)1/3
=
a1/3
b1/6
a1/6b2/3
ˆ Subtract the powers:
a1/3−1/6
b1/6−2/3
= a1/6
b−1/2
ˆ Back to radicals:
6
√
a
√
b
Practice Problems (for seminar)
Simplify the expressions
1. 3
1+a
− 1 · 3
2−a
− 1
2. 1
(a−b)2 − 1
(a−b)2+4ab
÷ ab
a2−b2
3. (x2−9)
x2+5x+6
· x+2
x−3
4. (x2
y−3
) · (x−1
y4
)
5. x2−x(x−2)−4
x2−4
6. Simplify: √
x − 2
x − 4
(Hint: factor the denominator and express as a conjugate if needed)
7. Rationalize the denominator:
5
2 −
√
3
BPM MATH0 – Week 1
Algebraic Expressions and Basic Equations
Solve the equations
8. x2
+ 2x + 1 = 0
9. 2x2
− 2x + 1 = x2
− 2x
10. (x + 1)2
= 2x + 5