MCR3U Polynomials Exam Review PDF

MCR3U Exam Review 1 Polynomials A polynomial is an algebraic expression with real coefficients and non-negative integer exponents. A polynomial with 1 term is called a monomial,. A polynomial with 2 te...

MCR3U Exam Review 1 Polynomials A polynomial is an algebraic expression with real coefficients and non-negative integer exponents. A polynomial with 1 term is called a monomial,. A polynomial with 2 terms is called a binomial,. A polynomial with 3 terms is called a trinomial,. The degree of the polynomial is determined by the value of the highest exponent of the variable in the polynomial. e.g. , degree is 2. For polynomials with one variable, if the degree is 0, then it is called a constant. If the degree is 1, then it is called linear. If the degree is 2, then it is called quadratic. If the degree is 3, then it is called cubic. We can add and subtract polynomials by collecting like terms. e.g. Simplify. The negative in front of the brackets applies to every term inside the brackets. That is, you multiply each term by –1. To multiply polynomials, multiply each term in the first polynomial by each term in the second. e.g. Expand and simplify. Factoring Polynomials To expand means to write a product of polynomials as a sum or a difference of terms. To factor means to write a sum or a difference of terms as a product of polynomials. Factoring is the inverse operation of expanding. Expanding   Factoring Sum or Product of difference of polynomials terms MCR3U Exam Review 2 Types of factoring: Common Factors: factors that are common among each term. e.g. Factor, Each term is divisible by. Factor by grouping: group terms to help in the factoring process. 1+6x+9x2 is a perfect e.g. Factor, Group 4mx – 4nx and square trinomial ny – my, factor each group Difference of squares Recall n – m = –(m – n) Common factor Factoring Find the product of ac. Find two numbers that multiply to ac and add to b. e.g. Factor, Product = 14 = 2(7) Product = 3(–6) = –18 = –9(2) Sum = 9 = 2 + 7 Sum = – 7 = –9 + 2 Decompose middle term –7xy into –9xy + 2xy. Factor by grouping. Sometimes polynomials can be factored using special patterns. Perfect square trinomial or e.g. Factor, Difference of squares e.g. Factor, Things to think about when factoring:  Is there a common factor?  Can I factor by grouping?  Are there any special patterns?  Check, can I factor ?  Check, can I factor ? MCR3U Exam Review 3 Rational Expressions For polynomials F and G, a rational expression is formed when. e.g. Simplifying Rational Expressions e.g. Simplify and state the restrictions. Factor the numerator and denominator. Note the restrictions. Simplify. State the restrictions. Multiplying and Dividing Rational Expressions e.g. Simplify and state the restrictions. Factor. Factor. Note restrictions. Note restrictions. Invert and multiply. Simplify. Note any new restrictions. State restrictions. Simplify. State restrictions. Adding and Subtracting Rational Expressions e.g. Simplify and state the restrictions. Factor. Factor. Note restrictions. Note restrictions. Simplify if possible. Simplify if possible. Find LCD. Find LCD. Write all terms Write all terms using LCD. using LCD. Add. Subtract. State restrictions. State restrictions. Note that after addition or subtraction it may be possible to factor the numerator and simplify the expression further. Always reduce the answer to lowest terms. MCR3U Exam Review 4 Radicals e.g. , is called the radical sign, n is the index of the radical, and a is called the radicand. is said to be a radical of order 2. is a radical of order 3. Like radicals: Unlike radicals: Same order, like radicands Entire radicals: Different order Different radicands Mixed radicals: A radical in simplest form meets the following conditions: MCR3U Exam Review 5 For a radical of order The radicand contains The radicand contains The index of a radical n, the radicand has no no fractions. no factors with must be as small as factor that is the nth 3 3 2 negative exponents. possible. power of an integer.   1 2 2 2 a1  a 6  1 a 22   a a 6  a 22  Simplest a2 6 form Simplest  a form 2  a  MCR3U Exam Review 6 Not simplest form Simplest form Simplest form Addition and Subtraction of Radicals To add or subtract radicals, you add or subtract the coefficients of each radical. e.g. Simplify. Express each radical in simplest form. Collect like radicals. Add and subtract. Multiplying Radicals e.g. Simplify. Use the distributive property to expand Multiply coefficients together. Multiply radicands together. Collect like terms. Express in simplest form. MCR3U Exam Review 7 Conjugates Opposite signs are called conjugates. Same terms Same terms When conjugates are multiplied the result is a rational expression (no radicals). e.g. Find the product. Dividing Radicals e.g. Simplify. Prime Factorization e.g. 180 Factor a number into its prime 3 60 factors using the tree diagram 6 10 method. 2 3 2 5 Exponent Rules Rule Description Example Product Quotient Power of a power Power of a quotient Zero as an exponent Negative exponents Rational Exponents MCR3U Exam Review 8 e.g. Evaluate. e.g. Simplify. Follow the order Power of a quotient. of operations. Evaluate brackets first. Power of a product. Solving Exponential Equations e.g. Solve for x. Add 8 to both sides. When the bases are Simplify. the same, equate the exponents. Note LS and RS are powers of Solve for x. 9, so rewrite them as powers using the same base. Don’t forget to check your solution! Functions A relation is a relationship between two sets. Relations can be described using: an equation an arrow diagram a graph a table g 2 x y 8 -1 1 2 in words 0 7 2 3 “output is three more than input” 6 -3 3 4 3 -5 -2 4 3 a set of ordered pairs function notation The domain of a relation is the set of possible input values (x values). The range is the set of possible output values (y values). e.g. State the domain and range. A: B: 4 Looking at the graph we C: Domain = {0, 1, 4} can see that y does not go What value of x will Range = {2, 3, 8} below 0. Thus, 2 make x – 5 = 0? x = 5 Domain = R The radicand cannot be less Range = than zero, so Domain = Range = MCR3U Exam Review 9 A function is a special type of relation in which every element of the domain corresponds to exactly one element of the range. and are examples of functions. is not a function because for every value of x there are two values of y. The vertical line test is used to determine if a graph of a relation is a function. If a vertical line can be passed along the entire length of the graph and it never touches more than one point at a time, then the relation is a function. e.g. A: 4 B: 4 This passes The line passes through the vertical more than one point, so this 2 line test, so 2 relation fails the vertical it is a line test. It is not a function. function. Inverse Functions The inverse, , of a relation, , maps each output of the original relation back onto the corresponding input value. The domain of the inverse is the range of the function, and the range of the inverse is the domain of the function. That is, if , then. The graph of is the reflection of the graph in the line. e.g. Given. Evaluate. Evaluate You want to find the value of Replace all the expression. You x’s with –3. are not solving for. Evaluate. Determine. Evaluate Rewrite as If you have not already determined do so. Interchange x and y. Solve for y. Using , replace all x’s with 2. Evaluate. 4 4 4 1 y  f ( x) e.g. Sketch the graph of the inverse of the given function 2 2. 2 Draw the Reflect the line y = x. graph in the line y = x. y  f (x) -2 -2 -2 -4 -4 -4 MCR3U Exam Review 10 The inverse of a function is not necessarily going to be a function. If you would like the inverse to also be a function, you may have to restrict the domain or range of the original function. For the example above, the inverse will only be a function if we restrict the domain to or. Transformations of Functions To graph from the graph consider: a – determines the vertical stretch. The graph is stretched vertically by a factor of a. If a < 0 then the graph is reflected in the x-axis, as well. k – determines the horizontal stretch. The graph is stretched horizontally by a factor of. If k < 0 then the graph is also reflected in the y-axis. p – determines the horizontal translation. If p > 0 the graph shifts to the right by p units. If p < 0 then the graph shifts left by p units. q – determines the vertical translation. If q > 0 the graph shifts up by q units. If q < 0 then the graph shifts down by q units. 4 When applying transformations to a graph the stretches and reflections should be applied before any translations. 2 e.g. The graph of is transformed into -2. Describe the transformations. -4 First, factor inside the 4 4 Shift to the Shift up by 1. brackets to determine the right by 2. 2 values of k and p. 2 There is a vertical stretch of This is the graph of -2 -2 3. -4 -4 A horizontal stretch of. The graph will be shifted 2 units to the right. e.g. Given the graph of sketch the graph of 4 4 2 2 Stretch vertically Reflect in y-axis. by a factor of 2. -2 -2 -4 -4  4 a>0 Quadratic Functions 2 minimum The graph of the quadratic function, , is a parabola. -5 5 When the parabola opens up. When the parabola opens down. maximum -2 a 0. The period of is , k > 0. The value of b determines the horizontal translation, known as the phase shift. The value of d determines the vertical translation. is the equation of the axis of the curve. e.g. e.g. gx = cos2x+1 2 1 1.5 0.5 1 gx = 0.5sinx+45 0.5 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 -0.5 -0.5 fx = sinx -1 fx = cosx -1