Polynomials & Factoring formulas

All 20 formulas in this topic

Difference of Squaresa2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)Instantly factor binomials that are differences of perfect squares by recognizing the pattern and splitting into a product of a sum and difference.Perfect Square Trinomiala2±2ab+b2=(a±b)2a^2 \pm 2ab + b^2 = (a \pm b)^2Quickly factor trinomials of the form a^2 \pm 2ab + b^2 by recognizing the perfect square pattern that gives (a \pm b)^2.Sum of Cubesa3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)Sum of Cubes is a factoring formula that breaks a binomial with two cubic terms into a product of a linear and a quadratic factor.Difference of Cubesa3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)Difference of Cubes is a factoring pattern that breaks down any cubic binomial subtraction into a linear and a quadratic factor.Factoring Trinomialsx2+bx+c=(x+p)(x+q),p+q=b,;pq=cx^2 + bx + c = (x + p)(x + q), \quad p + q = b, ; pq = cBreak a trinomial into two binomials by finding two numbers that add to b and multiply to c; use when solving equations or simplifying.Factoring by Groupingax+ay+bx+by=(a+b)(x+y)ax + ay + bx + by = (a + b)(x + y)Factoring by grouping factors four-term polynomials by pairing terms and extracting common factors—use it when you cannot factor out a GCF from all terms.Greatest Common Factor of a Polynomialab+ac=a(b+c)ab + ac = a(b + c)Factor a polynomial by identifying and pulling out the single largest expression that divides evenly into every term in the polynomial.Binomial Theorem(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}The Binomial Theorem expands (a+b)^n into a sum of terms, giving you a formula for computing powers of binomials without multiplying them out repeatedly.Pascal's Triangle(nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}Pascal's Triangle displays binomial coefficients and generates the coefficients for polynomial expansion, perfect for computing combinations quickly.Polynomial Long DivisionP(x)D(x)=Q(x)+R(x)D(x)\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}Polynomial long division breaks a fraction into quotient and remainder, letting you simplify rational expressions or test polynomial divisibility.Synthetic DivisionP(x)÷(x−c)P(x) \div (x - c)A fast method for dividing a polynomial by a binomial of the form (x - c), giving quotient and remainder efficiently without long division.Remainder TheoremP(x)=(x−c)Q(x)+P(c)P(x) = (x - c)Q(x) + P(c)The Remainder Theorem lets you find the remainder when a polynomial is divided by (x - c) by simply evaluating the polynomial at c.Factor Theorem(x−c)∣P(x)  ⟺  P(c)=0(x - c) \mid P(x) \iff P(c) = 0The Factor Theorem tells you when a linear binomial divides a polynomial: (x - c) is a factor of P(x) exactly when P(c) = 0.Rational Root Theoremx=±pq,p∣a0,;q∣anx = \pm\frac{p}{q}, \quad p \mid a_0, ; q \mid a_nThe Rational Root Theorem tells you which rational numbers could possibly be roots of a polynomial with integer coefficients.Fundamental Theorem of Algebradeg⁡P=n  ⟹  P has n complex roots\deg P = n \implies P \text{ has } n \text{ complex roots}Every polynomial equation of degree n has exactly n complex roots (counting multiplicity) — this guarantees solutions always exist.End Behavior of PolynomialsP(x)∼anxn as x→±∞P(x) \sim a_n x^{n} \text{ as } x \to \pm\inftyDescribes how a polynomial function behaves as x approaches positive or negative infinity, determined by its degree and leading coefficient.Multiplicity of a RootP(x)=(x−r)mQ(x),Q(r)≠0P(x) = (x - r)^m Q(x), \quad Q(r) \neq 0Multiplicity describes how many times a root appears as a factor, determining whether a polynomial graph touches or crosses the axis.Vertical Asymptotex=a where Q(a)=0,;P(a)≠0x = a \text{ where } Q(a) = 0, ; P(a) \neq 0Find where a rational function approaches infinity by identifying values where the denominator equals zero but the numerator doesn't.Horizontal Asymptotey=anbm when deg⁡P=deg⁡Qy = \frac{a_n}{b_m} \text{ when } \deg P = \deg QFind the horizontal asymptote of a rational function by comparing the degrees and leading coefficients of its numerator and denominator polynomials.Complex Numbersi2=−1,(a+bi)(c+di)=(ac−bd)+(ad+bc)ii^2 = -1, \quad (a + bi)(c + di) = (ac - bd) + (ad + bc)iSolve equations with no real solutions and multiply combinations of real and imaginary numbers using the complex number system.

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