Types of polynomials — by degree and terms, with factoring strategies

Question

Classify $3x^4 - 2x^2 + 7$ by degree and number of terms. What factoring strategy works for each polynomial type?

(CBSE 9 & 10 — polynomials chapter)

Solution — Step by Step

Why This Works

graph TD
    A["Factor a polynomial"] --> B{"Common factor?"}
    B -->|"Yes"| C["Factor out GCF first"]
    B -->|"No"| D{"Degree?"}
    C --> D
    D -->|"2 (quadratic)"| E{"Type?"}
    D -->|"3 (cubic)"| F["Try factor theorem<br/>or grouping"]
    D -->|"4 (biquadratic)"| G["Substitute t = x²<br/>Solve as quadratic in t"]
    E -->|"a² - b²"| H["(a+b)(a-b)"]
    E -->|"Trinomial"| I["Split middle term<br/>or quadratic formula"]
    E -->|"Perfect square"| J["(a ± b)²"]

Classification guides strategy. A quadratic trinomial calls for middle-term splitting. A biquadratic with only even powers calls for substitution. Knowing the type saves time by pointing you to the right technique immediately.

Alternative Method — Factor Theorem

For any polynomial $p(x)$: if $p(a) = 0$, then $(x - a)$ is a factor. Try small integer values ($0, \pm 1, \pm 2$) to find roots. This works for all degrees but is most practical for cubics and quartics.

Common Mistake