Polynomials
Grades 9–11 Algebra Polynomials
A polynomial is a sum of terms like
4x³ − 2x² + 7x − 5. Each term is a coefficient times x
to a power. The highest power is the degree —
this one's degree 3 (a "cubic").
- degree 0 → constant (7)
- degree 1 → linear (3x + 2)
- degree 2 → quadratic (x² − 5x)
- degree 3 → cubic
Combine like terms only
Like terms share the same variable and power.
3x² + 2x² combine; 3x² + 2x don't — squares
and plain x's are different species.
(4x² + 3x − 1) + (2x² − 5x + 6)
= (4x² + 2x²) + (3x − 5x) + (−1 + 6) ← group like terms
= 6x² − 2x + 5
Distributing
3(2x + 5) = 6x + 15 ← multiply everything inside
x(x + 4) = x² + 4x
Watch out
Subtracting a polynomial flips every sign inside:
(5x + 3) − (2x − 4) = 5x + 3 − 2x + 4 = 3x + 7. The −4 becomes +4.
Practice
The degree of 7x⁴ − 2x² + 9 is:
Combine: 5x + 3x
Simplify: (5x + 6) − (−2x + 4) (write like 7x+2)
Distribute: 3(2x + 5)
Next: factoring · Back: square roots