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1.6 - Solving Polynomial Equations MCB4U - Santowski (A) Review • To restate the Factor Theorem, if (ax - b) is a factor of P(x), then P(b/a) = 0. • A root or a zero is the x value (b/a) that makes the value of the polynomial zero. They have special graphical significance as the x-intercepts (i.e. that is the x value when the function has a value of zero) • So as an example, if x - 1 is a factor of x3 – 2x2 - 2 + 2, then P(1) = 0. The other way to state the same idea is that for P(x) = x3 – 2x2 - 2 + 2, then x - 1 is factor and that x = 1 is root of P(x) or that one x-intercept of the function is at x = 1. (B) Rational Root Theorem • Our previous observations (although limited in development) led to the following theorem: • Given that P(x) = anxn + an-1xn-1 + ….. + a1x1 + a0, if P(x) = 0 has a rational root of the form a/b and a/b is in lowest terms, then a must be a divisor of a0 and b must be a divisor of an (C) Rational Root Theorem • • • • • • • So what does this theorem mean? If we want to factor the polynomial P(x) = 2x3 – 5x2 + 22x – 10, then we first need to find a value a/b such that P(a/b) = 0 So the factors of the leading coefficient are {+1,+2} which are then the possible values for a The factors of the constant term, -10, are {+1,+2,+5,+10} which are then the possible values for b Thus the possible ratios a/b which we can test using the Factor Theorem are {+1,+½ ,+2,+5/2,+5,+10} As it then turns out, P(½) turns out to give P(x) = 0, meaning that x – ½ (or 2x – 1) is a factor of P(x) From this point on, we can then do the synthetic division (using ½) to find the quotient and then possibly other factor(s) of P(x) (C) Rational Root Theorem Example (D) Examples • ex.1. Solve 2x3 – 9x2 - 8x = -15 and then show on a GDC • • • Now graph both g(x) = 2x3 – 9x2 - 8x and then h(x) = -15 and find intersection • • Then graph: f(x) = 2x3 – 9x2 - 8x + 15 (D) Examples • Solve 2x3 + 14x - 20 = 9x2 - 5 and then show on a GDC • Explain that different solution sets are possible depending on the number set being used (real or complex) (D) Examples • ex. 3 Solve 2x4 - 3x3 + 2x2 - 6x - 4 = 0 then graph using roots, points, end behaviour. Approximate turning points, max/min points, and intervals of increase and decrease. • ex 4. The roots of a polynomial are 2, -3, 3 - 2i. The graph passes through (1, -64). Determine the equation of the polynomial and sketch. (E) Examples - Applications • ex 5. You have a sheet of paper 30 cm long by 20 cm wide. You cut out the 4 corners as squares and then fold the remaining four sides to make an open top box. – (a) Find the equation that represents the formula for the volume of the box. – (b) Find the volume if the squares cut out were each 2 cm by 2 cm. – (c) What are the dimensions of the squares that need to be removed if the volume is to be 1008 cm3? (E) Examples - Applications • The volume of a rectangular-based prism is given by the formula V(x) = -8x + x3 – 5x2 + 12 – (i) Express the height, width, depth of the prism in terms of x – (ii) State any restrictions for x. Justify your choice – (iii) what would be the dimensions on a box having a volume of 650 cubic units? – (iv) now use graphing technology to generate a reasonable graph for V(x). Justify your window/view settings (E) Examples - Applications • The equation p(m) = 6m5 – 15m4 – 10m3 + 30m2 + 10 relates the production level, p, in thousands of units as a function of the number of months of labour since October, m. • Use graphing technology to graph the function and determine the following: – maximums and minimums. Interpret in context – Intervals of increase and decrease. Interpret – Explain why it might be realistic to restrict the domain. Explain and justify a domain restriction – Would 0<m<3 be a realistic domain restriction? • Find when the production level is 15,500 units (try this one algebraically as well) (E) Examples - Applications • • • • • • • Use GDC to create a scatter-plot Use GDC to create and validate linear, quadratic, cubic and quartic regression eqns Discuss domain restrictions in each model Predict populations in 2006, 2016 What is the best regression model? Why? When will the pop. be 35,000,000 According to the quartic and cubic model, when was the population less than 25,000,000 • year Population in ‘000s 1911 7,207 1941 11,507 1961 18,238 1971 21,568 1981 24,820 1986 26,101 1991 28,031 1996 29,672 2001 30,755 (F) Internet Links • Finding Zeroes of Polynomials from WTAMU • Finding Zeroes of Polynomials Tutorial #2 from WTAMU • Solving Polynomials from Purple Math (G) Polynomials in Nested Form • An optional factoring technique that may make it easier for evaluating a polynomial • • • • Let P(x) = 2x3 – 3x2 + 5x – 7 Then P(x) = (2x2 – 3x + 5)x – 7 And P(x) = ((2x – 3)x + 5)x – 7 And P(x) = (((2)x – 3)x + 5)x – 7 • • • • So P(4) = (((2)4 – 3)4 + 5)4 – 7 P(4) = ((8 - 3)4 + 5)4 – 7 P(4) = (20 + 5)4 – 7 P(4) = 100 – 7 = 93 • OR P(4) = 2 x 4 = 8 - 3 = 5 x 4 = 20 +5 = 25 x 4 = 100 - 7 = 93 (H) Homework • Nelson text, page 60, Q1,2,8,9 on the first day. Graph Q8ac,9ac. • Nelson text page 61, Q11,12,13,15,19,22,23 on the second day as we focus on applications of polynomial functions