Factoring and roots Definition: A polynomial is a function of the form: n n−1 f(x) = anx + an−1x + ::: + a1x + a0 where an; an−1; : : : ; a1; a0 are real numbers and n is a nonnegative integer. The domain of a polynomial is the set of all real numbers. The degree of the polynomial is the largest power of x that appears. Division Algorithm for Polynomials: If p(x) and d(x) denote polynomial functions and if d(x) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q(x) and r(x) such that p(x) r(x) d(x) = q(x) + d(x) or p(x) = q(x)d(x) + r(x). where r(x) is either the zero polynomial or a polynomial of degree less than that of d(x) In the equation above, p(x) is the dividend, d(x) is the divisor, q(x) is the quotient and r(x) is the remainder. We say d(x) divides p(x) () the remainder is 0 () p(x) = d(x)q(x) () d(x) is a factor of p(x). p(x) If d(x) is a factor of p(x), the other factor of p(x) is q(x), the quotient of d(x) . p(x) • Given d(x) , divide (using Long Division or Synthetic Division (if applicable)) to get the quotient q(x) and remainder r(x). Write the answer in division algorithm form: p(x) = d(x)q(x) + r(x). x3+1 • Write x−1 in division law form. x3+1 2 2 3 2 x−1 = x + x + 1 + x−1 () x + 1 = (x − 1)(x + x + 1) + 2 3x3+4x2+x+7 • Write x2+1 in division law form. 3x3+4x2+x+7 −2x+3 x2+1 = 3x + 4 + x2+1 () 3x3 + 4x2 + x + 7 = (3x + 4)(x2 + 1) − 2x + 3 Remainder Theorem: Let f(x) be a polynomial function. If f(x) is divided by x − c, then the remainder is f(c). x-intercept: The number c is a root or zero of p(x) () p(c) = 0. So the x-intercepts are the roots of the polynomial. Factor Theorem: The number c is a root of p(x) () (x − c) is a factor of p(x). To find all the roots of p(x), completely factor p(x). 1 Theorem: If a > 0; x2 − a = (x − p(a))(x + p(a)). *But x2 + a has no roots and can't be factored anymore. Example: Find all the roots and factor x3 + 5x2 + 8x + 4 given that −1 is a root. (x − a) = (x − (−1)) = (x + 1) so we divide by (x + 1). x3+5x2+8x+4 2 2 x+1 = x + 4x + 4 = (x + 2)(x + 2) = (x + 2) So, x3 + 5x2 + 8x + 4 = (x + 1)(x + 2)2 The roots of the polynomial are the x such that f(x) = 0: (x + 1)(x + 2)2 = 0 () x = −2 or x = −1 () Roots:−2; −1. Example: Find all the roots and factor x3 − x2 − 2x + 2 given that 1 is a root. p p x3−x2−2x+2 2 x−1 = x − 2 = (x + 2)(x − 2). p p So x3 − x2 − 2x + 2 = (x − 1)(x + 2)(x − 2). p p ) Roots:− 2; 1; 2. Example: Find all the roots and factor 2x2 + 2x − 2. Factor out the coefficient of x2 and then find the roots using the quadratic formula: 2x2 + 2x − 2 = 2(x2 + x − 1) p p p p −b± b2−4ac −1± 12−4(1)(−1) −1− 5 −1+ 5 Roots: x = 2a = 2(1) = 2 , 2 p p 2 −1− 5 −1+ 5 Factorization: 2x + 2x − 2 = 2(x − 2 )(x − 2 ) Rational Zeros Theorem: Let f be a polynomial function of degree 1 or nigher of the form n f(x) = anx + ··· + a1x + a0, an 6= 0, a0 6= 0 p where each coefficient is an integer. Then all the possible rational zeros are of the form q where p is a factor of a0 and q is a factor of an. Example: List the potential rational zeros of f(x) = 2x3 + 11x2 − 7x − 6. p : ±1; ±2; ±3; ±6 q : ±1; ±2; Thus, p 1 3 q : ±1; ±2; ±3; ±6; ± 2 ; ± 2 . Definition: If (x − c)m is a factor of a polynomial f and (x − c)m+1 is not a factor of f, then c is called a zero of multiplicity m of f. 2 Graphs of Polynomial Functions Definition: A polynomial function is a function of the form: n n−1 f(x) = anx + an−1x + ::: + a1x + a0 • The degree of a polynomial is the largest power of x that appears in the function. n • The leading term of the polynomial above is anx . We call an the leading coefficient. • The y-intercept of the polynomial above is the constant term ao. Polynomials are among the simplest expressions in algebra. The graph of every polynomial function is both smooth and continuous. To say a function is smooth means the graph contains no sharp corners or cusps. For a function to be continuous means that the graph has no gaps or holes. Example: f(x) = −3x4 + x2 − 5 has leading term = −3x4, leading coefficient = −3, degree = 4, and constant term = −5. Example: f(x) = x − x3 has leading term = −x3, leading coefficient = −1, degree = 3, and constant term = 0. Important Facts:2 Math 140 Lecture 9 y= x y= x y= -x2 See inside text’s front cover for area and volume formulas. n • For large x (near ±∞), the graph looks like the graph of the leading term anx . `A 6'x6' tarp forms the top of a pup tent. Write the height h of a pup tent as a function of the floor area A. • As x goes to 1, y goes to +1 if a > 0, to −∞ if a < 0. • Graphs of polynomials of odd degree go to +1 in one direction, and go to −∞ in the other. Consider the graphs of f(x) = x3 and f(x) = −x3: 3 y= x4 h 3 6 y= x3 y= -x b/2 b/2 2 b 2 2 Given: A 6b, h 2 3 Want A in h. Need b in h. b 2 32 h2, b 9 h2 , b 29 h2 2 2 For large x (near +5), graph looks like the leading term axn. 2 2 v A 6b 629 h 12 9 h . • Graphs of polynomials of even degree either go to +1 in both directions or to −∞ in v As x goes to 5, y goes to +5 if a > 0, to -5 if a < 0. both directions . Consider the graphs of f(x) = x2 and f(x) = −x2: (right circular cylinder) v Graphs of odd degree go to +5 in one direction, -5 in `The height of aMath can 140 Lecture is 9 three times 2 2 y= x 3 3 y= x y= -x the radius.See inside text’s front cover for area and volume formulas. the other, like y = x , y = -x . r `A 6'x6' tarp formsS = curved the surfacetop of areaa pup tent. Write the v Graphs of even degree either go to +5 in both h 2 2 height h of a pup tent as a function of the floor area A. directions or to -5 in both directions, like y = x , y = -x . Use the factored form to get the roots and their degrees (the degree of a root is the exponent of its factor). 3 y= x4 h h At roots of degree 1, the graph crosses x-axis like y = x or y = -x. 6 v 3 3 2p r v At roots of odd degreesUse the 3, factored 5, 7, ..., they= form xgraph of acrosses polynomial they= x -axis-x to find the roots and their degrees (the degree of a b/2 b/2 3 root3 is the exponent of its factor). 2 b 2 2 like y = x or y = -x . Given:Given: h A = 3r6, b S, h= 2prh2 3 v At roots of even degrees, 2, 4, 6, ..., the graph touches but doesn’t • At2 roots of2 degree 1, the graph crosses the x-axis like y = x or y = −x. (a)Want Express A in the h. 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