Title NMAT Quantitative Part 2.pdf ✅

Polynomial Operations ADDITION AND SUBTRACTION: Adding and subtracting polynomials isthe same as the procedure used in combining like terms. When adding polynomials, simply drop the parenthesis and combine like terms. When subtracting polynomials, distribute the negative first, then combine like terms. Examples: Addition: (2x2 +3x -7)+(3x2 -4x -10) = 2x2 +3x2 +3x -4x -7 -10 = 5x 2 -x -17 Subtraction: (5x 2 -12x +1)-(2x2 +3x -7) = 5x 2 -12x +1 -2x2 -3x +7 = 3x2 -15x +8 MULTIPLICATION: 1. Monomial times Monomial: To multiply a monomial times a monomial, just multiply the numbers then multiply the variables using the rulesfor exponents. Example: (-2x2 y)(5xy7 ) = -2 · 5x 2 · x · y · y 7 = -10x3 y 8 2. Monomial times Polynomial: Simply use the distributive property to multiply a monomial times a polynomial. Examples: a. -2x(x 2 + 3x - 8) = -2x(x 2 ) - 2x(3x) - 2x(-8) = -2x3 + 6x2 + 16x b. 5x 2 (-2x4 + 3y - 6) = 5x 2 (-2x4 ) + 5x 2 (3y) + 5x 2 (-6) = -10x6 + 15x 2 y - 30x2 3. Binomial times a Binomial: To multiply two binomials, use the FOIL method (First times first, Outside times outside, Inside timesinside, and Last times last). Example: (x + 2)(x -3) = x(x) + x(-3) + 2(x) + 2(-3) = x 2 -3x + 2x -6 = x 2 - x -6 Special Products: The following formulas may be used in these special cases as a short cut to the FOIL method. Difference of Squares: (a + b)(a -b) = a 2 -b 2 Perfect Squares: (a + b) 2 = a 2 + 2ab + b 2 Example: (3x+4)(3x -4) = 9x2 -16 Example: (x + 4) 2 = x 2 + 2(x)(4) + 4 2 = x 2 + 8x + 16 (a -b) 2 = a 2 -2ab +b 2 Example: (x -3) 2 = x 2 -2(x)(3) + 3 2 = x 2 -6x + 9
2 4. Polynomial times polynomial: To multiply two polynomials where at least one has more than two terms, distribute each term in the first polynomial to each term in the second. Examples: a. (x 2 +3x -4)(x 2 -6x +5) = x 2 (x 2 ) + x 2 (-6x) + x 2 (5) +3x(x 2 ) +3x(-6x)+ 3x(5)-4(x 2 )-4(-6x)-4(5) = x 4 -6x3 +5x 2 +3x3 -18x2 +15x -4x2 +24x -20 =x 4 -3x3 -17x2 +39x -20 b. (2x -3)(4x2 -5x +1) = 2x(4x2 ) +2x(-5x)+2x(1)-3(4x2 )-3(-5x)-3(1) =8x3 -10x2 +2x -12x2 +15x -3 = 8x3 + 22x2 + 17x -3 DIVISION: 1. Division by Monomial: Each term of the polynomial is divided by the monomial and it issimplified asindividual fractions. Examples: 2. Division by Binomial or Larger Polynomial: Use the long division format asfollows: • Both the divisor and the dividend must be written in descending order. • Any missing powers should be replaced by zero. • All remainders are in fraction form (remainder/divisor) and are added to the quotient. Examples: a. (x 2 - 2x - 15) ÷ (x + 3) = x -5 9x 3 -x+3 S b. = 3x + 2x + 1 + 3x-2 3x-2 x + 3 x − 5 x 2 − 2x −15 3x 2 + 2x +1 -(x 2 +3x) -5x-15 -(-5x-15) 0 3x − 2 9x 3 + 0x 2 − x + 3 -(9x3 -6x 2 ) 6x 2 -x -(6x2 -4x) 3x + 3 -(3x -2) 5 -
The following steps can be used as a general guideline for approaching math word problems. Steps Tips for success Example 1. Read the problem over once to become familiar with the scenario. If necessary, read the problem twice or more. A certain recipe requires 2 1 cups of 2 flour, half a cup of sugar, three apples and 1 cup of cranberries. This recipe makes 12 muffins. What amount of sugar is required if you want to make exactly 15 2. Understand the problem situation. Connect the situation to your personal experience and/or real life. muffins? 3. Specify exactly what you are trying to find. Look for the question sentence in the problem. What amount of sugar is required if you want to make exactly 15 muffins? 4. Underline the information that you think is important for solving the question. Sometimes quantities are written out in words and not as numbers! half a cup of sugar This recipe makes 12 muffins. 5. Write down the important information (the givens) in point form. 1 cup sugar for 12 2 muffins 6. Mathematize the situation described - How are the givens Since there is a and come up with a way to solve for related to one another comparison of the unknown using the givens. mathematically? - Look for “clue words.” different quantities, a proportion can be used to solve for the unknown. 1 : 12 = x : 15 7. Solve for the unknown. There may be more than one step! 1/2 = x 12 15 12x = 1 (15) 2 x = 15÷ 12 2 x = 5 8 8. Interpret your final answer. Does the final answer seem reasonable? 5 is greater than 1 . This 8 2 makes sense since we are making more muffins than the original recipe. 9. Communicate the final result to the reader. - Using the question sentence to help you write a sentence that answers this question. - Include units if applicable. 5 cups of sugar is 8 required to make 15 muffins according to the recipe.