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