Page 130 - The Ontario Curriculum, Grades 11 and 12: Mathematics, 2007

Grade 12, College Preparation
3. Solving Problems Involving Polynomial Equations
2. Connecting Graphs and Equations of Polynomial Functions
THE ONTARIO CURRICULUM, GRADES 11 AND 12 | Mathematics
polynomial function that models the relation- ship between height above the ground and time for a falling object)
Sample problem: The forces acting on a hori- zontal support beam in a house cause it to sag by d centimetres, x metres from one end of the beam. The relationship between d and x can be represented by the polynomial function
d(x) = 1 x(1000 – 20x2 + x3). Graph the 1850
function, using technology, and determine the domain over which the function models the relationship between d and x. Determine the length of the beam using the graph, and explain your reasoning.
By the end of this course, students will:
2.1 factor polynomial expressions in one variable, of degree no higher than four, by selecting and applying strategies (i.e., common factor- ing, difference of squares, trinomial factoring)
Sample problem: Factor: x4 – 16; x3 – 2x2 – 8x.
2.2 make connections, through investigation using graphing technology (e.g., dynamic geometry software), between a polynomial function given in factored form [e.g.,
f(x) = x(x – 1)(x + 1)] and the x-intercepts of its graph, and sketch the graph of a polyno- mial function given in factored form using its key features (e.g., by determining intercepts and end behaviour; by locating positive and negative regions using test values between and on either side of the x-intercepts)
Sample problem: Sketch the graphs of
f(x) = –(x – 1)(x + 2)(x – 4) and
g(x) = –(x – 1)(x + 2)(x + 2) and compare their shapes and the number of x-intercepts.
2.3 determine, through investigation using tech- nology (e.g., graphing calculator, computer algebra systems), and describe the connection
between the real roots of a polynomial equa- tion and the x-intercepts of the graph of the corresponding polynomial function [e.g., the real roots of the equation x4 – 13x2 + 36 = 0 are the x-intercepts of the graph of
f(x)=x4 –13x2+36]
Sample problem: Describe the relationship between the x-intercepts of the graphs of linear and quadratic functions and the real
roots of the corresponding equations. Inves- tigate, using technology, whether this rela- tionship exists for polynomial functions of higher degree.
By the end of this course, students will:
3.1 solve polynomial equations in one variable, of degree no higher than four (e.g., x2 – 4x = 0, x4 – 16 = 0, 3x2 + 5x + 2 = 0), by selecting and applying strategies (i.e., common factoring; difference of squares; trinomial factoring), and verify solutions using technology (e.g., using computer algebra systems to determine the roots of the equation; using graphing technol- ogy to determine the x-intercepts of the corresponding polynomial function)
Sample problem: Solve x3 – 2x2 – 8x = 0.
3.2 solve problems algebraically that involve polynomial functions and equations of degree no higher than four, including those arising from real-world applications
3.3 identify and explain the roles of constants and variables in a given formula (e.g., a constant can refer to a known initial value or a known fixed rate; a variable changes with varying conditions)
Sample problem: The formula P = P0 + kh is used to determine the pressure, P kilopascals, at a depth of h metres under water, where
k kilopascals per metre is the rate of change of the pressure as the depth increases, and
P0 kilopascals is the pressure at the surface. Identify and describe the roles of P, P0 , k, and h in this relationship, and explain your reasoning.
3.4 expand and simplify polynomial expressions involving more than one variable [e.g., sim- plify – 2xy(3x2y3 – 5x3y2 )], including expres- sions arising from real-world applications
Sample problem: Expand and simplify the expression π(R + r)(R – r) to explain why it represents the area of a ring. Draw a diagram of the ring and identify R and r.
3.5 solve equations of the form xn = a using rational exponents (e.g., solve x3 = 7 by raising both sides to the exponent 31 )
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