Polynomials

Concept

A polynomial is a sum of terms aₙxⁿ + … + a₁x + a₀. The degree is the highest exponent — it caps the number of real zeros.

Factor theorem — (x − a) is a factor of P(x) if and only if P(a) = 0.
End behaviour — driven by the leading term. Even degree, both ends move together (up if leading coefficient is positive). Odd degree, ends go in opposite directions.
Common factoring tools — pull out the GCF, recognise difference of squares a² − b² = (a − b)(a + b), perfect-square trinomials, and grouping.

Solution

GCF. x³ − 4x = x(x² − 4)

Factor. x² − 4 is a difference of squares: x(x − 2)(x + 2)

Zeros: x = 0, 2, −2. Three real zeros — the maximum possible for a degree-3 polynomial.

Solution

Degree. Highest exponent is 4 — a degree-4 polynomial. (Even degree.)

Leading coeff. Coefficient of x⁴ is 2 — positive.

End behaviour. Even degree + positive leading coefficient ⇒ both ends rise to +∞.

Degree 4, leading coeff 2; the graph rises on the far left and far right.

8 questions on degree, end behaviour, factoring, the factor theorem, and zeros.

Practice test · 8 questions Question 1 of 8 · Score 0