Classifying Polynomials
Identify: Is 5x a monomial, binomial, or trinomial?
5x is a monomial.
Add: (2x + 3) + (x − 1)
(2x+3)+(x−1) = 3x + 2
Subtract: (3x + 5) − (x + 2)
(3x+5) − (x+2) = 2x + 3
Multiply: x(3x + 2)
x(3x+2) = 3x2 + 2x
Classify: How many terms and what degree is 6x?
Linear Monomial
Identify the degree and number of terms: 3x2 + 7x
Quadratic Binomial
Add: (4x2 + x) + (3x2 − 2x)
(4x2+x)+(3x2−2x) = 7x2 − x
Subtract: (4x2 + 2x) − (x2 + x + 1)
(4x2+2x) − (x2+x+1) = 3x2 + x − 1
Multiply: 2x( x2 + 3 )
2x(x2+3) = 2x3 + 6x
Add then Classify: Add (x + 2) + (2x − 5). Then say whether the result is a monomial, binomial, or trinomial.
(x+2)+(2x−5) = 3x − 3 = Binomial
Identify: 4x3 − 2x + 1 — classify by number of terms and give degree.
Cubic Trinomial
Add: (5xy + 2x) + (−2xy + 4x)
(5xy+2x)+(−2xy+4x) = 3xy + 6x
Subtract: (5xy − 2x) − (3xy + x)
(5xy−2x) − (3xy+x) = 2xy − 3x
Multiply (FOIL): (x + 2)(x + 3)
(x+2)(x+3) = x2 + 5x + 6
Multiply then Simplify: x(2x + 3) + (x + 1). First multiply, then add like terms.
x(2x+3) + (x+1) = 2x2 + 3x + x + 1 = 2x2 + 4x + 1
True/False: x2y is a polynomial with one variable. Explain briefly.
False — x2y has two variables (x and y); it's a polynomial with two variables.
Add: (3x2 − 2x + 5) + (−x2 + 4x − 3)
(3x2−2x+5)+(−x2+4x−3) = 2x2 + 2x + 2
Subtract: (x3 + 4x2 − x) − (2x3 − x2 + 3x)
(x3+4x2−x) − (2x3−x2+3x) = −x3 + 5x2 − 4x
Multiply: (x − 4)(x + 5)
(x−4)(x+5) = x2 + x − 20
Subtract then Classify: (4x2 + x − 3) − (x2 + 2x + 1). State the degree of the result.
(4x2 + x − 3) − (x2 + 2x + 1) = 3x2 − x − 4 = degree 2 (Quadratic)
Identify: Name the polynomial type and degree for
2x4 − 5x2 + 3x − 7.
Quartic Polynomial
Add: (2x3 + x2 − x) + (−x3 + 3x − 4)
(2x3+x2−x)+(−x3+3x−4) = x3 + x2 + 2x − 4
Subtract: (3x2 − 5x + 2) − (−x2 + 4x − 1)
(3x2−5x+2) − (−x2+4x−1) = 4x2 − 9x + 3
Multiply (with two variables): (2x − y)(x + 3y)
(2x−y)(x+3y) = 2x2 + 5xy − 3y2
Mixed multi-step: (1) Multiply (x + 2)(x − 3). (2) Add 4x to your result. (3) Classify the final polynomial (number of terms and degree).
(x+2)(x−3) = x2 − 3x + 2. Add 4x: x2 + x + 2. Final: Quadratic Trinomial