What are the 10 formulas for Factorization?
Answers
1 Factoring FormulasFor any real numbers a and b,(a + b)2 = a2 + 2ab + b2 Square of a Sum(a − b)2 = a2 − 2ab + b2 Square of a Differencea2 − b2 = (a − b)(a + b) Difference of Squaresa3 − b3 = (a − b)(a2 + ab + b2) Difference of Cubesa3 + b3 = (a + b)(a2 − ab + b2) Sum of Cubes2 Exponentiation RulesFor any real numbers a and b, and any rational numbers pqand rs,ap/qar/s = ap/q+r/s Product Rule= aps+qrqsap/qar/s = ap/q−r/s Quotient Rule= aps−qrqs(ap/q)r/s = aYpr/qs Power of a Power Rule(ab)p/q = ap/qbp/q Power of a Product Ruleabp/q=ap/qbp/q Power of a Quotient Rulea0 = 1 Zero Exponenta−p/q =1ap/q Negative Exponents1a−p/q = ap/q Negative ExponentsRemember, there are different notations:√q a = a1/q√qap = ap/q = (a1/q)p3 Quadratic FormulaFinally, the quadratic formula: if a, b and c are real numbers, then the quadratic polynomialequationax2 + bx + c = 0 (3.1)has (either one or two) solutionsx =−b ±√b2 − 4ac2a(3.2)4 Points and LinesGiven two points in the plane,P = (x1, y1), Q = (x2, y2)you can obtain the following information:1. The distance between them, d(P, Q) = p(x2 − x1)2 + (y2 − y1)2.2. The coordinates of the midpoint between them, M =x1 + x22,y1 + y22.3. The slope of the line through them, m =y2 − y1x2 − x1=riserun.Lines can be represented in three different ways:Standard Form ax + by = cSlope-Intercept Form y = mx + bPoint-Slope Form y − y1 = m(x − x1)where a, b, c are real numbers, m is the slope, b (different from the standard form b) is the y-intercept,and (x1, y1) is any fixed point on the line.5 CirclesA circle, sometimes denoted J, is by definition the set of all points X := (x, y) a fixed distance r,called the radius, from another given point C = (h, k), called the center of the circle,K def = {X | d(X, C) = r} (5.1)Using the distance formula and the square root property, d(X, C) = r ⇐⇒ d(X, C)2 = r2, we seethat this is preciselyK def = {(x, y) | (x − h)2 + (y − k)2 = r2} (5.2)which gives the familiar equation for a circle.FunctionsIf A and B are subsets of the real numbers R and f : A → B is a function, then the average rateof change of f as x varies between x1 and x2 is the quotientaverage rate of change = ∆y∆x=y2 − y1x2 − x1=f(x2) − f(x1)x2 − x1(6.1)It’s a linear approximation of the behavior of f between the points x1 and x2.7 Quadratic FunctionsThe quadratic function (aka the parabola function or the square function)f(x) = ax2 + bx + c (7.1)can always be written in the formf(x) = a(x − h)2 + k (7.2)where V = (h, k) is the coordinate of the vertex of the parabola, and furtherV = (h, k) = −b2a, f −b2a (7.3)That is h = −b2aand k = f(−b2a).8 Polynomial DivisionHere are the theorems you need to know:Theorem 8.1 (Division Algorithm) Let p(x) and d(x) be any two nonzero real polynomials.There there exist unique polynomials q(x) and r(x) such thatp(x) = d(x)q(x) + r(x)orp(x)d(x)= q(x) + r(x)d(x)where 0 ≤ deg(r(x)) < deg(d(x))Here p(x) is called the dividend, d(x) the divisor, q(x) the quotient, and r(x) the remainder. Theorem 8.2 (Rational Zeros Theorem) Let f(x) = anx2 + an−1xn−1 + · · · + a1x + a0 be areal polynomial with integer coefficients ai (that is ai ∈ Z). If a rational number p/q is a root, orzero, of f(x), thenp divides a0 and q divides anTheorem 8.3 (Intermediate Value Theorem) Let f(x) be a real polynomial. If there are realnumbers a < b such that f(a) and f(b) have opposite signs, i.e. one of the following holdsf(a) < 0 < f(b)f(a) > 0 > f(b)then there is at least one number c, a < c < b, such that f(c) = 0. That is, f(x) has a root in theinterval (a, b). Theorem 8.4 (Remainder Theorem) If a real polynomial p(x) is divided by (x − c) with theresult thatp(x) = (x − c)q(x) + r(r is a number, i.e. a degree 0 polynomial, by the division algorithm mentioned above), thenr = p(c) 9 Exponential and Logarithmic FunctionsFirst, the all important correspondencey = ax ⇐⇒ loga(y) = x (9.1)which is merely a statement that ax and loga(y) are inverses of each other.Then, we have the rules these functions obey: For all real numbers x and yax+y = axay(9.2)ax−y =axay(9.3)a0 = 1 (9.4)and for all positive real numbers M and Nloga(MN) = loga(M) + loga(N) (9.5)logaMN= loga(M) − loga(N) (9.6)loga(1) = 0 (9.7)loga(MN ) = N loga(M) (9.8)
all 9 formulas are there hope it will help it was to diffcult to type this all so pls its a request that pls mark me as a brainlist .