Example: (3x + 1)(7x + 10) 3x and 7x are the first pair of terms, 3x and 10 are the outermost pair of terms, 1 and 7x are the innermost pair of terms, and 1 and 10 are the last pair of terms. Therefore, (3x)(7x) + (3x)(10) + (1)(7x) + (1)(10) = 21x 2 + 30x + 7x + 10. After we combine like terms, we are left with the answer: 21x 2 + 37x + 10. Factoring Factoring is the reverse of multiplication: 2(x + y) = 2x + 2y Multiplication 2x + 2y = 2(x + y) Factoring THREE BASIC TYPES OF FACTORING ■ Factoring out a common monomial: 10x 2 – 5x = 5x(2x – 1) and xy – zy = y(x – z) ■ Factoring a quadratic trinomial using the reverse of FOIL: y 2 – y – 12 = (y – 4) (y – +3) and z 2 – 2z + 1 = (z – 1)(z – 1) = (z – 1) 2 ■ Factoring the difference between two squares using the rule: a 2 – b 2 = (a + b)(a – b) and x 2 – 25 = (x + 5)(x – 5) REMOVING A COMMON FACTOR If a polynomial contains terms that have common fac- tors, the polynomial can be factored by using the reverse of the distributive law. Example: In the binomial 49x 3 + 21x,7x is the greatest common factor of both terms. Therefore, you can divide 49x 3 + 21x by 7x to get the other factor. ᎏ 49x 3 7 + x 21x ᎏ = ᎏ 4 7 9 x x 3 ᎏ + ᎏ 2 7 1 x x ᎏ = 7x 2 + 3 Thus, factoring 49x 3 + 21x results in 7x(7x 2 + 3). I SOLATING VARIABLES USING FRACTIONS It may be necessary to use factoring in order to isolate a variable in an equation. Example: If ax – c = bx + d, what is x in terms of a, b, c, and d? 1. The first step is to get the x terms on the same side of the equation. ax – bx = c + d 2. Now you can factor out the common x term on the left side. x(a – b) = c + d 3. To finish, divide both sides by a – b to isolate the variable of x. ᎏ x( a a – – b b) ᎏ = ᎏ c a + – d b ᎏ 4. The a – b binomial cancels out on the left, result- ing in the answer: x = ᎏ c a + – d b ᎏ Quadratic Trinomials A quadratic trinomial contains an x 2 term as well as an x term. x 2 – 5x + 6 is an example of a quadratic trino- mial. It can be factored by reversing the FOIL method. ■ Start by looking at the last term in the trinomial, the number 6. Ask yourself, “What two integers, when multiplied together, have a product of posi- tive 6?” ■ Make a mental list of these integers: 1 × 6 –1 × –6 2 × 3 –2 × –3 ■ Next, look at the middle term of the trinomial, in this case, the –5x. Choose the two factors from the above list that also add up to –5. Those two factors are: –2 and –3 ■ Thus, the trinomial x 2 – 5x + 6 can be factored as (x – 3)(x – 2). –THE SAT MATH SECTION– 122 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 122 ■ Be sure to use the FOIL method to double-check your answer: (x – 3)(x – 2) = x 2 – 5x + 6 The answer is correct. Algebraic Fractions Algebraic fractions are very similar to fractions in arithmetic. Example: Write ᎏ 5 x ᎏ – ᎏ 1 x 0 ᎏ as a single fraction. Just like in arithmetic, you need to find the LCD of 5 and 10, which is 10. Then change each frac- tion into an equivalent fraction that has 10 as a denominator. ᎏ 5 x ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 5 x( ( 2 2 ) ) ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 1 2 0 x ᎏ – ᎏ 1 x 0 ᎏ = ᎏ 1 x 0 ᎏ Reciprocal Rules There are special rules for the sum and difference of reciprocals. Memorizing this formula might make you more efficient when taking the SAT. ■ If x and y are not 0, then ᎏ 1 x ᎏ + 1y = ᎏ x x + y y ᎏ ■ If x and y are not 0, then ᎏ 1 x ᎏ – ᎏ 1 y ᎏ = ᎏ y x – y x ᎏ Quadratic Equations A quadratic equation is an equation in which the greatest exponent of the variable is 2, as in x 2 + 2x – 15 = 0. A quadratic equation has two roots, which can be found by breaking down the quadratic equation into two simple equations. Zero-Product Rule The zero - product rule states that if the product of two or more numbers is 0, then at least one of the numbers is 0. Example: (x + 5)(x – 3) = 0 Using the zero-product rule, it can be deter- mined that either x + 5 = 0 or that x – 3 = 0. x + 5 = 0 or x – 3 = 0 –5 –5 + 3 +3 x = –5 or x = +3 Thus, the possible values of x are –5 and 3. Solving Quadratic Equations by Factoring ■ If a quadratic equation is not equal to zero, you need to rewrite it. Example: Given x 2 – 5x = 14, you will need to subtract 14 from both sides to form x 2 – 5x – 14 = 0. This quadratic equation can now be factored using the zero-product rule. ■ It may be necessary to factor a quadratic equation before solving it and to use the zero-product rule. Example: x 2 + 4x = 0 must first be factored before it can be solved: x(x + 4). Graphs of Quadratic Equations The (x,y) solutions to quadratic equations can be plot- ted. It is important to look at the equation at hand and to be able to understand the calculations that are being performed on every value that gets substituted into the equation. For example, below is the graph of y = x 2 . x y 1234567 1 2 3 4 5 –1 –2 –3 –1–2–3–4–5–6–7 –THE SAT MATH SECTION– 123 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 123 For every number you put into the equation (as an x value), you know that you will simply square the number to get the corresponding y value. The SAT may ask you to compare the graph of y = x 2 with the graph of y = (x – 1) 2 . Think about what hap- pens when you put numbers (your x values) into this equation. If you have an x = 2, the number that gets squared is 1. The graph will look identical to the y = x 2 graph, except it will be shifted to the right by 1: How would the graph of y = x 2 compare with the graph of y = x 2 – 1? In this case, you are still squaring your x value, and then subtracting 1. This means that the whole graph of y = x 2 has been moved down 1 point. Rational Equations and Inequalities Recall that rational numbers are all numbers that can be written as fractions ( ᎏ 2 3 ᎏ ), terminating decimals (.75), and repeating decimals (.666 . . . ). Keeping this in mind, it’s no surprise that rational equations are just equations in fraction form. Rational inequalities are also in fraction form and involve the symbols <, >, ≤, and ≥ instead of an equals sign. Example: Given ᎏ (x + x 5 2 ) + (x x 2 – – x 20 –12) ᎏ = 10, find the value of x. Factor the top and bottom: = 10 Note that you can cancel out the (x + 5) and the (x – 4) terms from the top and bottom to yield: x + 3 = 10 Thus, x = 7. Radical Equations Some algebraic equations on the SAT will include the square root of the unknown. The first step is to isolate the radical. When you have accomplished this, you can then square both sides of the equation to solve for the unknown. Example: 4͙b ෆ + 11 = 27 To isolate the variable, subtract 11 from both sides: 4͙b ෆ = 16 Next, divide both sides by 4: ͙b ෆ = 4 Last, square both sides: ͙b 2 ෆ = 4 2 b = 16 (x + 5)(x + 3)(x – 4) ᎏᎏᎏ (x + 5)(x – 4) x y 1234567 1 2 3 4 5 –1 –2 –3 –1–2–3–4–5–6–7 x y 1234567 1 2 3 4 5 –1 –2 –3 –1–2–3–4–5–6–7 –THE SAT MATH SECTION– 124 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 124 Sequences Involving Exponential Growth When analyzing a sequence, you always want to try and find the mathematical operation that you can per- form to get the next number in the sequence. Look carefully at the sequence: 2,6,18,54, You probably noticed that each successive term is found by multiplying the prior term by 3; (2 × 3 = 6, 6 × 3 = 18, and so on.) Since we are multiplying each term by a constant number, there is a constant ratio between the terms. Sequences that have a constant ratio between terms are called geometric sequences. On the SAT, you may, for example, be asked to find the thirtieth term of a geometric sequence like the one above. There is not enough time for you to actu- ally write out all the terms, so you should notice the pattern: 2,6,18,36, Term 1 = 2 Term 2 = 6, which is 2 × 3 Term 3 = 18, which is 2 × 3 × 3 Term 4 = 54, which is 2 × 3 × 3 × 3 Another way of looking at this, would be to use exponents: Term 1 = 2 Term 2 = 2 × 3 1 Term 3 = 2 × 3 2 Term 4 = 2 × 3 3 So, if the SAT asks you for the thirtieth term, you know that term 30 = 2 × 3 29 . Systems of Equations A system of equations is a set of two or more equations with the same solution. Two methods for solving a system of equations are substitution and linear combination. Substitution Substitution involves solving for one variable in terms of another and then substituting that expression into the second equation. Example: 2p + q = 11 and p + 2q = 13 1. First, choose an equation and rewrite it, isolating one variable in terms of the other. It does not matter which variable you choose. 2p + q = 11 becomes q = 11 – 2p 2. Second, substitute 11 – 2p for q in the other equation and solve: p + 2(11 – 2p)= 13 p + 22 – 4p = 13 22 – 3p = 13 22 = 13 + 3p 9= 3p p = 3 3. Now substitute this answer into either original equation for p to find q. 2p + q = 11 2(3) + q = 11 6 + q = 11 q = 5 4. Thus, p = 3 and q = 5. Linear Combination Linear combination involves writing one equation over another and then adding or subtracting the like terms so that one letter is eliminated. Example: x – 9 = 2y and x + 3 = 5y 1. Rewrite each equation in the same form. x – 9 = 2y becomes x – 2y = 9 and x + 3 = 5y becomes x – 5y = 3 –THE SAT MATH SECTION– 125 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 125 2. If you subtract the two equations, the x terms will be eliminated, leaving only one variable: Subtract: x – 2y = 9 –(x – 5y = 3) ᎏ 3 3 y ᎏ = ᎏ 6 3 ᎏ y = 2 is the answer. 3. Substitute 2 for y in one of the original equations and solve for x: x – 2y = 9 x – 2(2) = 9 x – 4 = 9 x – 4 + 4 = 9 + 4 x = 13 4. The answer to the system of equations is y = 2 and x = 13. Functions, Domain, and Range Functions are written in the form beginning with: f(x) = For example, consider the function f(x) = 3x – 8. If you are asked to find f(5), you simply substitute the 5 into the given function equation. f(x) = 3x – 8 becomes f(5) = 3(5) – 8 f(5) = 15 – 8 = 7 In order to be classified as a function, the function in question must pass the vertical line test. The verti- cal line test simply means that a vertical line drawn through a graph of the function in question CANNOT pass through more than one point of the graph. If the function in question passes this test, then it is indeed a function. If it fails the vertical line test, then it is NOT a function. All of the x values of a function, collectively, are called its domain. Sometimes, there are x values that are outside of the domain, and these are the x values for which the function is not defined. All of the solutions to f(x) are collectively called the range. Values that f(x) cannot equal are said to be outside of the range. The x values are known as the independent vari- ables. The outcome of the function depends on the x val- ues, so the y values are called the dependent variables. Qualitative Behavior of Graphs and Functions In addition to being able to solve for f(x) and make judgments regarding the range and domain, you should also be able to analyze the graph of a function and inter- pret, qualitatively, something about the function itself. Look at the x-axis, and see what value for f(x) cor- responds to each x value. For example, consider the portion of the graph shown below. For how many values does f(x) = 3? When f(x) = 3, the y value (use the y-axis) will equal 3. As shown below, there are five such points. x y 1234567 1 2 3 4 5 –1 –2 –3 –1–2–3–4–5–6–7 x y 1234567 1 2 3 4 5 –1 –2 –3 –1–2–3–4–5–6–7 –THE SAT MATH SECTION– 126 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 126 Geometry Review To begin this section, it is helpful to become familiar with the vocabulary used in geometry. The list below defines some of the main geometrical terms. It is followed by an overview of geometrical equations and figures. arc part of a circumference area the space inside a two-dimensional figure bisect cut in two equal parts circumference the distance around a circle chord a line segment that goes through a circle, with its endpoint on the circle congruent identical in shape and size. The geometric notation of “congruent” is ≅ . diameter a chord that goes directly through the center of a circle—the longest line you can draw in a circle equidistant exactly in the middle hypotenuse the longest leg of a right triangle, always opposite the right angle line a straight path that continues infinitely in two directions. The geometric notation for a line is AB ឈ . line segment the part of a line between (and including) two points. The geometric notation for a line segment is PQ . parallel lines in the same plane that will never intersect perimeter the distance around a figure perpendicular two lines that intersect to form 90-degree angles quadrilateral any four-sided figure radius a line from the center of a circle to a point on the circle (half of the diameter) ray a line with an endpoint that continues infinitely in one direction. The geometric notation for a ray is AB . tangent line a line meeting a smooth curve (such as a circle) at a single point without cutting across the curve. Note that a line tangent to a circle at point P will always be perpendicular to the radius drawn to point P. volume the space inside a three-dimensional figure –THE SAT MATH SECTION– 127 5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 127 . you can divide 49x 3 + 21x by 7x to get the other factor. ᎏ 49x 3 7 + x 21x ᎏ = ᎏ 4 7 9 x x 3 ᎏ + ᎏ 2 7 1 x x ᎏ = 7x 2 + 3 Thus, factoring 49x 3 + 21x results in 7x(7x 2 + 3). I SOLATING VARIABLES. common fac- tors, the polynomial can be factored by using the reverse of the distributive law. Example: In the binomial 49x 3 + 21x,7x is the greatest common factor of both terms. Therefore, you. Example: (3x + 1)(7x + 10) 3x and 7x are the first pair of terms, 3x and 10 are the outermost pair of terms, 1 and 7x are the innermost pair of terms, and 1 and 10 are the last pair