BPM MATH0 – Week 5
Polynomial and Power Functions
Week 5 – Polynomial and Power Functions
Weekly Goals
ˆ Apply algebraic techniques to manipulate polynomial expressions.
ˆ Use binomial identities and expansion patterns to simplify expressions.
ˆ Divide polynomials using long division and interpret the remainder.
ˆ Complete the square to rewrite quadratic functions in vertex form.
Solved Examples – With Detailed Steps
Example 1: Expand the expression: (x + 2)3
Steps:
ˆ Use the binomial pattern: (a + b)3
= a3
+ 3a2
b + 3ab2
+ b3
ˆ Here: a = x, b = 2
ˆ Result:
x3
+ 3x2
· 2 + 3x · 4 + 8 = x3
+ 6x2
+ 12x + 8
Example 2: Divide with remainder: (2x3
+ 3x2
− x + 5) ÷ (x − 2)
Steps:
ˆ Use polynomial long division: first, divide the leading terms: 2x3
x
= 2x2
ˆ Multiply (x − 2) by 2x2
and subtract:
(2x3
+ 3x2
) − (2x3
− 4x2
) = 7x2
ˆ Bring down the next term: −x, and repeat:
7x2
x
= 7x, then (7x2
− x) − (7x2
− 14x) = 13x
ˆ Bring down the final term: +5:
13x
x
= 13, then (13x + 5) − (13x − 26) = 31
ˆ Final result:
2x2
+ 7x + 13 with remainder 31
ˆ Final form:
2x2
+ 7x + 13 +
31
x − 2
BPM MATH0 – Week 5
Polynomial and Power Functions
Example 3: Rewrite x2
+ 6x + 5 in vertex form
Steps:
ˆ Complete the square:
x2
+ 6x + (9 − 9) + 5 = (x2
+ 6x + 9) − 9 + 5 = (x + 3)2
− 4
ˆ Final form: (x + 3)2
− 4
Example 4: Sketch the graph of the function and find its vertex:
f(x) = x2
− 4x + 3
Steps:
ˆ Complete the square: f(x) = (x2
− 4x + 4) − 4 + 3 = (x − 2)2
− 1
ˆ Vertex: (2, −1), opens upward
ˆ Axis of symmetry: x = 2
ˆ Intercepts: f(0) = 3, solve x2
− 4x + 3 = 0 ⇒ x = 1, 3
Practice Problems for Seminar
Algebraic Manipulations
1. Expand: (x − 1)4
2. Factor: x3
+ 3x2
− x − 3
Division and Remainders
1. Divide: 3x3
− x2
+ 2x + 7 by x + 1
2. Divide and write as mixed expression: x2+1
x+2
BPM MATH0 – Week 5
Polynomial and Power Functions
Vertex Form and Completing the Square
1. Rewrite in vertex form: x2
− 10x + 27
2. Find the vertex of f(x) = −2x2
+ 4x + 5 using completing the square
Sketch
3. f(x) = −x2
+ 4x − 3
4. g(x) = x3
− 3x
5. h(x) = 1
x2
6. k(x) = x2/3