Polynomials, Quadratics & Functions

1 · Multiplying binomials

Multiply each term of the first by each term of the second, then combine. (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.

2 · Special products

(a + b)² = a² + 2ab + b², so (x + 4)² = x² + 8x + 16, not x² + 16. The middle term is the one people forget.

(a − b)(a + b) = a² − b², so (x − 6)(x + 6) = x² − 36: the middle terms cancel.

3 · Factoring

For x² + bx + c, find two numbers that multiply to c and add to b. x² + 7x + 12: 3 and 4 work, so (x + 3)(x + 4).

Take out a common factor first: 3x² − 12 = 3(x² − 4) = 3(x − 2)(x + 2).

4 · Solving quadratics

Get 0 on one side, factor, and set each factor to 0. x² − x − 20 = 0 → (x − 5)(x + 4) = 0 → x = 5 or x = −4. The signs flip from the factors.

Do not divide by x: x² = 6x has two solutions, x = 0 and x = 6. Moving 6x over and factoring, x(x − 6) = 0, keeps both.

5 · Using identities

Some questions never need x and y themselves. If x + y = 9 and x − y = 3, then x² − y² = (x + y)(x − y) = 27.

If x − y = 5 and xy = 14, then x² + y² = (x − y)² + 2xy = 25 + 28 = 53.

6 · Parabolas

In y = a(x − h)² + k, the vertex is (h, k). y = (x − 2)² + 3 has vertex (2, 3), and since a square is never negative, its smallest value is 3.

If a is negative the parabola opens downward and the vertex is the highest point. Watch the sign: y = (x + 2)² means h = −2.

7 · Function notation

f(x) = x² − 3x is a rule; f(−2) means "put −2 in for x." Use parentheses: f(−2) = (−2)² − 3(−2) = 4 + 6 = 10.

8 · Composition

f(g(x)) means do g first, then f. With f(x) = 2x + 1 and g(x) = x²: f(g(3)) = f(9) = 19, but g(f(3)) = g(7) = 49. The order matters.

9 · Reading a table

If y changes by the same amount each time x goes up by 1, the rule is a line: the change is the slope. If the changes themselves grow by the same amount each time, the rule has an x².

10 · Growth and decay

Doubling repeats a multiplication. 200 bacteria that double every 3 hours: 12 hours is 4 doublings, so 200 × 2⁴ = 3,200. Multiplying 200 by 2 × 4 gives only 1,600.

Halving works the same way: 800 mg that halves every 6 hours leaves 800 ÷ 2⁴ = 50 mg after 24 hours.