Completely Factored Polynomials
{"Image DocumentName": "Completely Factored Polynomials", "Image ImageDescription": "Polynomial Factor Theorem. f(x) = 2(x4 + 7x3 + 9x2 \u2013 27x \u2013 54) x \u2013 k = x \u2013 (\u20133) \u2013 \u201327 \u201354. \u20133. \u2013 \u20133. \u201318. (x + 3) is a factor. Completely Factored Polynomials. We show how the lead coefficient fits in with all of the factors to produce a completely factored polynomial. A polynomial is completely factored if and only if it can be written as the product of a constant and one or more linear factors. This includes the possibility that two or more of zeros of the polynomial are complex numbers (which occur in complex conjugate pairs). If we insist that only real numbers be considered, then the pairs of complex conjugates form irreducible quadratic factors. By irreducible quadratics we mean quadratic factors that cannot be factored using only real numbers, e.g. x2 + x + 1. We first factor out the constant 2 from every term, leaving the lead coefficient as 1. Then, in the illustration, synthetic division is used to show that (x + 3) is a factor of the polynomial f(x), that is, that -3 is a zero for f(x), leaving a lower degree polynomial Q1. The Q1 is then divided by (x + 3) again, showing the there are two factors of (x + 3) with a remaining polynomial Q2 of lower degree than Q1. The process is repeated leaving finally a linear polynomial (x \u2013 2) as the last factor. We note that (x + 3) appears as a factor three times, i.e. (x + 3)3 is a factor of f(x). This is called a zero of multiplicity 3. Having already factored the constant, we can then write down the polynomial in completely factored form. \u2013 \u20133 \u201318. Q1. \u20133. \u2013 (x + 3) is a second factor \u20136. 7/23/2013. Polynomial Factors. Polynomial Factors. 7/23/2013.", "Image XResolution": "37", "Image YResolution": "37", "Image ResolutionUnit": "Pixels/Centimeter", "Image Software": "SlidePlayer", "Image YCbCrPositioning": "Centered"}