PRE-INTERMEDIATE
MATH
5.1 Polynomials
5.1 Introduction to Polynomials
What is a Polynomial?
A polynomial is defined as an expression that is
composed of variables, constants, and exponents, that are
combined using mathematical operations such as addition,
subtraction, multiplication and division (No division operation
by a variable).
Polynomial is made up of two terms, namely Poly (meaning
“many”) and Nomial (meaning “terms.”)
Standard Form of a Polynomial
P(x) = an x n + an−1 x n−1 + an−2 xn−2 +. . . +a1 x + a0
where an , an−1 , an−2 , . . . , a1 , a0 are called coefficients of x n , x n−1 , x n−2 , . . . , x and constant term
respectively and it should belong to real number (∈ ℝ).
Notation
The polynomial function is denoted by P(x) where x represents the variable. For example,
P(x) = x 2 + 5x + 11
If the variable is denoted by a, then the function will be P(a)
Degree of a Polynomial
The degree of a polynomial is defined as the highest exponent of a monomial within a
polynomial.
Polynomials can also be classified by the degree (largest exponent of the variable).
Polynomial
Degree
Example
Constant
P(x) = 6
0 degree (no power of x)
Linear
1st degree (x to the 1st power)
P(x) = 3x + 1
Quadratic
P(x) = 4𝑥 2 + 𝑥 + 1
2nd degree (𝑥 2 )
Cubic
3rd degree (𝑥 3 )
P(x) = 6𝑥 3 + 4𝑥 2 + 3𝑥 + 1
4
Quartic
4th degree (𝑥 )
P(x) = 6𝑥 4 + 3𝑥 3 + 3𝑥 2 + 2𝑥 + 1
Types of Polynomials
Polynomials can be classified (named) by the number of terms.
Polynomial
Number of terms
Monomial
1 term
Binomial
2 terms
Trinomial
3 terms
Example: Addition of Polynomials
Simplify (5x 2 + 4x + 4) + (2x 2 + 5x + 2)
Method 1: Add vertically
Line up like terms. Then add the coefficients.
5x 2 + 4x + 4
+ 2x 2 + 5x + 2
7x 2 + 9x + 6
Example: Subtraction of Polynomials
Simplify (−10x 2 − 4x + 7) − (x 2 − 9)
Method 1: Subtract vertically
Line up like terms. Then rewrite
subtraction as addition of the opposite.
(−10x 2 − 4x + 7) − (x 2 − 9)
−10x 2 − 4x + 7
+(−x 2
+ 9)
2
−11x − 4x + 16
Direction: Complete the table below.
Example/s
2𝑥 2 , 𝑥 , 3𝑦 , 29
x + 2 , 𝑥2 + 𝑥
𝑥 2 + 2𝑥 + 20
Method 2: Add horizontally
Group like terms. Then add the coefficients
(5x 2 + 4x + 4) + (2x 2 + 5x + 2)
2
= (5x + 2x 2 ) + (4x + 5x) + (4 + 2)
= 7x 2 + 9x + 6
Method 2: Subtract horizontally
Group like terms. Then rewrite subtraction as
addition of the opposite.
(−10x 2 − 4x + 7) − (x 2 − 9)
2
= (−10x − 4x + 7) + (−x 2 + 9)
= (−10x 2 −x 2 ) + (−4x) + (7 + 9)
Group like terms
Simplify
= −11x 2 − 4x + 16
PRE-INTERMEDIATE
1.
2.
3.
4.
5.
MATH
Polynomial
Standard form
Degree
3 − 7x − 9x 2
5 − 6x 3
−4
−10 + 5x
8x − 2 − 6x 3
−9x 2 − 7x + 3
2nd
Number of
Terms
3
Name
Quadratic trinomial
Directions: Add and subtract the polynomials. Write the answer in standard form.
6. (5x 2 + 8x − 10) + (12 − x + 3x 2 )
7. (8x − x 3 + 4 − 9x 2 ) + (7x 3 + 9x 2 − 10 − 8x)
8. (9x 2 − 4x + 8) − (12x 2 − x − 3)
9. (x − 2x 3 + 8 − 7x 2 ) − (−3x 3 + 5x + 8)
Practice: Adding and Subtracting Polynomials
1. (6x 2 − 7x + 1) + (x 2 + 3x − 2)
2. (3x 2 − 2x + 12) + (−5 + 2x + 2x 2 )
3. (5x 2 − x − 5) + (x 2 + 4x + 4)
4. (4x 2 − 4x + 8) + (x 2 + 3x + 6)
5. (2x 2 − 2x + 3) + (12 − x + 3x 2 )
6. (x 2 + 6x + 9) − (9 + 12x + 4x 2 )
7. (9x 2 + 12x + 4) − (10x 2 − 10x + 25)
8. (−9x 2 + 8x − 5) − (−4x 2 + 5x − 8)
9. (6x 2 + 13x + 11) − (2x 2 − 4x + 3)
10. (−3x 2 + 3x − 16) − (−15x 2 − 20x − 25)
11. State the degree of the monomial
9x 4
12. Find the product of 2x(3x 3 + 5x 2 + 5)