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MATH!!! EXAM PREP!!!!

MATH!!! EXAM PREP!!!!. ConoR RoweN. . If the same number is added to both sides of an equation, the two sides remain equal. if (x) = (y) , then ( x) + z = (y) + z. Addition Property (of Equality). Multiplication Property (of Equality).

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MATH!!! EXAM PREP!!!!

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  1. MATH!!! EXAM PREP!!!! ConoR RoweN

  2. . If the same number is added to both sides of an equation, the two sides remain equal. if (x) = (y), then (x) + z = (y) + z. Addition Property (of Equality) Multiplication Property (of Equality) For all real numbers  a  and  b , and for  c ≠ 0 , a = b     is equivalent to     ac = bc

  3. D = D Reflexive Property (of Equality) Symmetric Property (of Equality) if h = b, then b = h Transitive Property (of Equality) if a = b and b = c, then a = c .

  4. (a + b) + c = a + (b + c) Associative Property of Addition Associative Property of Multiplication When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4)

  5. Commutative Property of Addition When 2 numbers are added together, the sum is the same regardless of the order of the addends. example 4 + 2 = 2 + 4 Commutative Property of Multiplication When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4

  6. Distributive Property (of Multiplication over Addition)multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together. 4 × (2 + 3) = 4 × 2 + 4 × 3

  7. Prop of Opposites or Inverse Property of Addition EXAMPLE:: (8 - 8 = 0) Inverse Property of Addition---- Any number added to its opposite integer will always equal ZERO. If using addition the order of the numbers doesn’t matter [Ex. 3 + (-3) = 0 or (-3) + 3 = 0] Prop of Reciprocals or Inverse Prop. of Multiplication For every number, x, except zero, has a multiplicative inverse,1/x. EXAMPLE x * (1/x) = 1

  8. Identity Property of Addition the sum of zero and any number or variable is the number or variable itself. EXAMPLES:::: 4459907 + 0 = 4459907, - 1178 + 0 = - 1178, y + 0 = y are few examples illustrating the identity property of addition. Identity Property of Multiplication the product of 1 and any number or variable is the number or variable itself. EXAMPLES::::: 3 × 1 = 3, - 118997006 × 1 = - 118997006, ybbh × 1 = y bbh few examples illustrating the identity property of multiplication.

  9. Multiplicative Property of ZeroThe product of 0 and any number results in 0.That is, for any real number a, a × 0 = 0. Closure Property of Addition the sum of any two real numbers equals another real number. 2, 5 = real numbers. 2 + 5 = 7(real number). Closure Property of Multiplication the product of any two real numbers equals another real number. 4, 7 = real numbers. 4 × 7 = 28(real number)

  10. Product of Powers Propertyto multiply powers having the same base, add the exponents.That is, for a real number non-zero a and two integers m and n, am × an =am+n. Power of a Product Property find the power of each factor and then multiply Power of a Power Property the power of a power can be found by multiplying the exponents.for a non-zero real number a and two integers m and n, (am)n =amn.

  11. Quotient of Powers Property divide powers that have the same base, subtract the exponents.That is, for a non-zero real number a and two integers m and n,. Power of a Quotient Property the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. That is, for any two non-zero real numbers a and b and any integer m,  .

  12. Zero Power Propertyany variable with a zero exponent is equal to oneexample:: c°= 1 Negative Power Property Any variable with a negative exponent is equal to 1 over its reciprocal. ExAmPLE:: cˉⁿ = 1/cn

  13. Zero Product Property the product of two real numbers is zero, then at least one of the numbers in the product (factors) must be zero. Cbr = 0 THEN c = 0, b = 0, or r = 0

  14. Product of Roots Property for any nonnegative real numbers a and b.EXAmple::: √ab = √a • √b Quotient of Roots Property For any nonnegative real number a and any positive real number b: eXaMpLe √a/b = √a/√b

  15. Root of a Power PropertyI could not find this property anywhere. Power of a Root Property  the square root of anything to the 2nd power is that number. ExAmPLE!!!!!!!!!!!! square root of 121 is 11 and -11.

  16. Now you will take a quiz!Look at the sample problem and give the name of the property illustrated. 1. a + b = b + a Answer: Commutative Property (of Addition)

  17. Now you will take a quiz!Look at the sample problem and give the name of the property illustrated. 2. 4 + 2 = 2 + 4 Answer: Community Property of Addition

  18. 1St power inequalities

  19. Single Sign Inequality Problems X is less than or equal to 2 Written like: x≤2 Answer this! How is k less than or equal to 6 written? k≤6

  20. Conjunction Two sentences combined by the word and . EXAMPLE: x › 2and x ‹ 1

  21. Disjunction Two sentences combined by the word or. Example:: -3 ‹ x OR x ‹ 4

  22. In the next slides you will review:Linear equations in two variables

  23. SLOPES Positive Slope= line running upward (rising) from left to right Negative Slope= lines that are falling from left to right. Horizontal Slope= lines that run left to right (flat) with no slope (0=slope). Vertical Slope= Lines that rise up and down (undefined= slope)

  24. SLOPE FORMULAS! General= Ax + By + C= 0 Standard= Ax + By = C Slope= rise/run Slope Formula= Y2 - Y1/ X2 - X1 Slope-Intercept Form= y=mx + b (m=slope) (b= y-int.) Point-Slope= y – y1 = m ( x – x1)

  25. y x change in y 2 m = = 1 change in x (0,-4) (1, -2) Example: Graph the line y = 2x– 4. • The equation y = 2x– 4 is in the slope-intercept form. So, m = 2 and b = -4. 2. Plot the y-intercept, (0,-4). 3. slope= 2. 2 4. Start at the point (0,4). move 1 space to the right and 2 up to place the second point on the line. 1 (1,-2) is also on the line. 5. Draw the line through these points that you just made!

  26. 1 1 1 2 2 2 Example:The graph of the equation y – 3 = -(x – 4) is a line of slope m = - passing through the point (4,3). y m = - 8 (4, 3) 4 x 4 8 y–y1 = m(x – x1) is in point-slope form. slope (m) passes through the point (x1, y1).

  27. Linear SYSTEMS!

  28. Substitution Method Steps Solve one equation for one of the variables Substitute this expression in the other equation and solve for variable. Substitute this value in the equation in step 1 and solve Check values in both equations

  29. Addition/subtraction method Add or subtract the equations to eliminate one variable Solve the resulting equation for the other variable Substitute in either original equation to find the value of the first variable Check in both original equations

  30. Dependent, inconsistent, consistent

  31. FACTORING

  32. The simplest method of factoring a polynomial is to factor out the greatestcommon factor (GCF) of each term. Example: Factor 18x3+ 60x. 18x3 = 2 · 3 · 3 · x · x · x Factor each term. =(2 · 3 · x) · 3 · x · x 60x=2 · 2 · 3 · 5 · x =(2 · 3 · x) · 2 · 5 Find the GCF. GCF = 6x 18x3+ 60x=6x(3x2) + 6x(10) Apply the distributive law to factor the polynomial. = 6x(3x2 + 10) Check the answer by multiplication. 6x(3x2 + 10) = 6x(3x2) + 6x(10) = 18x3+ 60x

  33. Example: Factor 4x2 – 12x + 20. GCF = 4. = 4(x2 – 3x + 5) 4(x2 – 3x + 5) = 4x2 – 12x + 20 Check the answer. A common binomial factor can be factored out of certain expressions. Example: Factor the expression 5(x + 1) – y(x + 1). 5(x + 1) – y(x + 1) = (x + 1) (5 – y) (x + 1) (5 – y)= 5(x + 1) – y(x + 1) Check.

  34. difference of two squares Example: Factor x2 – 9y2. x2 – 9y2 = (x)2– (3y)2 Write terms as perfect squares. = (x + 3y)(x– 3y) . The same method can be used to factor any expression which can be written as a difference of squares. Example: Factor (x + 1)2 – 25y4. = (x + 1)2 – (5y2)2 (x + 1)2 – 25y4 = [(x + 1) + (5y2)][(x + 1) – (5y2)] = (x + 1 + 5y2)(x + 1 – 5y2)

  35. Some polynomials can be factored by grouping termsto produce a common binomial factor. Examples:1. Factor 2xy + 3y – 4x – 6. 2xy + 3y – 4x – 6 = (2xy + 3y) – (4x + 6) Group terms. = y (2x + 3) – 2(2x + 3) Factor each pair of terms. = (2x + 3)(y – 2) Factor out the common binomial. 2. Factor 2a2 +3bc– 2ab –3ac. 2a2 +3bc– 2ab –3ac =2a2– 2ab+3bc–3ac Rearrange terms. = (2a2– 2ab)+(3bc–3ac) Group terms. = 2a(a– b)+3c(b –a) Factor. = 2a(a– b)–3c(a –b) b – a = –(a – b). = (a–b)(2a – 3c) Factor.

  36. To factor a simple trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example, x2 + 10x + 24 = (x +4)(x +6). Factoring these trinomialsis based on reversing the FOIL process. Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b. Example: Factor x2 + 3x + 2. x2 + 3x + 2 = (x + a)(x + b) F O I L = x2 + bx + ax + ba Apply FOIL to multiply the binomials. = x2 + (b + a)x + ba Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2. = x2 + (1 +2)x + 1·2 Therefore, x2 + 3x + 2 = (x + 1)(x + 2).

  37. Negative Factors of 15 Sum -1, -15 -15 -3, -5 -8 Example: Factor x2 – 8x + 15. = (x+ a)(x+b) x2 – 8x + 15 = x2+ (a + b)x + ab and ab = 15. Therefore a + b = -8 It follows that both a and b are negative. x2 – 8x + 15 = (x – 3)(x – 5). Check: = x2 – 8x + 15. (x – 3)(x – 5) = x2– 5x– 3x + 15

  38. PositiveFactors of 36 Sum 1, 36 37 2, 18 20 3, 12 15 4, 9 13 6, 6 12 13 36 Example: Factor x2 + 13x + 36. x2 + 13x + 36 = (x+ a)(x+b) = x2+ (a + b)x + ab Therefore a and b are: two positive factors of 36 whose sum is 13. = (x + 4)(x+ 9) x2 + 13x + 36 Check: (x + 4)(x + 9) = x2 + 9x + 4x + 36 = x2 + 13x + 36.

  39. A polynomial is factored completely when it is written as a product of factors that can not be factored further. Example: Factor 4x3 – 40x2 + 100x. 4x3 – 40x2 + 100x The GCF is 4x. = 4x(x2 – 10x + 25) Use distributive property to factor out the GCF. = 4x(x – 5)(x – 5) Factor the trinomial. Check: 4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25) = 4x(x2 – 10x + 25) = 4x3 – 40x2 + 100x

  40. Factoring complex trinomials of the formax2+ bx + c, (a  1) can be done by decomposition or cross-check method. Example: Factor 3x2+8x +4. 3  4 = 12 Decomposition Method 1, 12 2, 6 3, 4 2. We need to find factors of12 1. Find the product of first and last terms is8 whose sum 3. Rewrite the middle term decomposed into the two numbers 3x2+2x + 6x +4 = (3x2+2x) + (6x +4) 4. Factor by grouping in pairs = x(3x +2) + 2(3x +2) = (3x + 2) (x + 2) 3x2+8x +4 = (3x + 2) (x + 2)

  41. Example: Factor 4x2 + 8x – 5. 4  5 = 20 We need to find factors of20 1, 20 2, 10 4, 5 whose difference is8 Rewrite the middle term decomposed into the two numbers 4x2–2x + 10x –5 = (4x2–2x) + (10x –5) = 2x(2x –1) + 5(2x –1) Factor by grouping in pairs = (2x –1) (2x +5) 4x2 + 8x – 5= (2x –1)(2x – 5)

  42. QUADRATIC Equations

  43. QUADRATICS • A binomial expression has just two terms (usually an x term and a constant). There is no equal sign.  Its general form is ax + b, where a and b are real numbers and a ≠ 0. • One way to multiply two binomials is to use the FOIL method. FOIL stands for the pairs of terms that are multiplied: First, Outside, Inside, Last. • This method works best when the two binomials are in standard form (by descending exponent, ending with the constant term). • The resulting expression usually has four terms before it is simplified. Quite often, the two middle (from the Outside and Inside) terms can be combined.

  44. For example:

  45. QUADRATICS • The opposite of multiplying two binomials is to factor or break down a polynomial (many termed) expression. • Several methods for factoring are given in the text. Be persistent in factoring! It is normal to try several pairs of factors, looking for the right ones. • The more you work with factoring, the easier it will be to find the correct factors. • Also, if you check your work by using the FOIL method, it is virtually impossible to get a factoring problem wrong.  • Remember!  When factoring, always take out any factor that is common to all the terms first.

  46. A quadratic equation involves a single variable with exponents no higher than 2. • Its general form is where a, b, and c are real numbers and .  • For a quadratic equation it is possible to have two unique solutions, two repeated solutions (the same number twice), or no real solutions. • The solutions may be rational or irrational numbers.

  47. To solve a quadratic equation, ONLY IF ITS factorable: •     1.  Make sure the equation is in the general form.  •     2.  Factor the equation. •     3.  Set each factor to zero.  •     4.  Solve each simple linear equation.

  48. To solve a quadratic equation if you can’t factor the equation: • Make sure the equation is in the general form.  • Identify a, b, and c. • Substitute a, b, and c into the quadratic formula: • Simplify.

  49. Tips & Tricks • The cool, easy thing about the quadratic formula is that it works on any quadratic equation when put in the form general form. • When having trouble factoring a problem, the quadratic formula might be quicker. • be sure and check your solution in the original quadratic equation.

  50. Rational Expressions

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