A polynomial is an expression made up of variables and coefficients combined using addition, subtraction, and multiplication, where every exponent on a variable is a non-negative integer; an example is \(4x^3 - 2x + 7\). The degree of a polynomial is the highest power of the variable that appears, so \(5x^4 + 3x^2 - 1\) has degree \(4\). Polynomials are added or subtracted by combining like terms, and they are multiplied by distributing each term of one polynomial across every term of the other. For two binomials, a convenient shortcut is the FOIL method (First, Outer, Inner, Last), which expands \((a + b)(c + d)\) into \(ac + ad + bc + bd\). Applied to \((2x + 3)(x - 4)\), FOIL gives \(2x^2 - 8x + 3x - 12\), which combines to \(2x^2 - 5x - 12\).
Factoring reverses the multiplication process and is one of the most useful skills in algebra. The first step in factoring any polynomial is to identify and pull out the greatest common factor, or GCF. For example, the GCF of \(6x^3 + 9x^2\) is \(3x^2\), and factoring it out gives \(3x^2(2x + 3)\). Several recognizable patterns make factoring faster: the difference of squares pattern \(a^2 - b^2 = (a + b)(a - b)\) applies to expressions like \(x^2 - 16 = (x + 4)(x - 4)\); perfect square trinomials follow \(a^2 + 2ab + b^2 = (a + b)^2\) or \(a^2 - 2ab + b^2 = (a - b)^2\), as in \(x^2 + 10x + 25 = (x + 5)^2\); and a trinomial of the form \(x^2 + bx + c\) factors as \((x + p)(x + q)\) where \(p\) and \(q\) are two numbers whose product is \(c\) and whose sum is \(b\). The sum and difference of cubes formulas, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) and \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), can be remembered with the mnemonic SOAP for the signs (Same, Opposite, Always Positive).
A quadratic equation is a polynomial equation of degree 2, written in standard form as \(ax^2 + bx + c = 0\) with \(a \neq 0\). The expression \(b^2 - 4ac\) beneath the square root in the quadratic formula is called the discriminant, and it reveals the nature of the solutions: if the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution (a repeated root); and if it is negative, there are two complex conjugate solutions. The quadratic formula itself provides the solutions to any quadratic equation in standard form. Applied to \(2x^2 + 3x - 2 = 0\), the discriminant is \(9 + 16 = 25\), and the formula gives \(x = \frac{-3 \pm 5}{4}\), producing the solutions \(x = \frac{1}{2}\) and \(x = -2\).
Two other important techniques for working with quadratics are completing the square and writing in vertex form. Completing the square rewrites a quadratic as \(a(x - h)^2 + k\) by taking half of the coefficient of \(x\), squaring it, and adding and subtracting that value. For example, to solve \(x^2 + 6x + 5 = 0\), you first rewrite it as \(x^2 + 6x = -5\) and add \((\frac{6}{2})^2 = 9\) to both sides to obtain \((x + 3)^2 = 4\); taking square roots then gives \(x + 3 = \pm 2\), so \(x = -1\) or \(x = -5\). The vertex form \(y = a(x - h)^2 + k\) has its vertex at the point \((h, k)\), and the sign of \(a\) tells you whether the parabola opens upward (if \(a > 0\)) or downward (if \(a < 0\)). By Vieta's formulas, the sum of the roots of \(ax^2 + bx + c = 0\) is \(-\frac{b}{a}\) and the product of the roots is \(\frac{c}{a}\), a quick way to check your answers. Factoring is often the simplest method when applicable, as with \(x^2 - 5x + 6 = 0\), which becomes \((x - 2)(x - 3) = 0\) and yields \(x = 2\) or \(x = 3\). The Remainder Theorem adds another tool: when a polynomial \(f(x)\) is divided by \((x - c)\), the remainder equals \(f(c)\), so if \(f(c) = 0\), then \((x - c)\) is a factor.