A Comprehensive Guide To Polynomial Operations, Coordinate Geometry, And Mathematical Simplification

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A Comprehensive Guide to Polynomial Operations, Coordinate Geometry, and MathematicalSimplificationPolynomials and Mathematical Simplification: A Step-by-Step Approach(10x + 12) + (6x + 10) add the polynomialLet's break down the givenexpression step by step:Given: (10x+12)+(6x+10)(10x + 12) + (6x + 10)Step 1: Remove the parenthesesSince there are no negative signs outside the parentheses, we can directly combine the terms inside theparentheses.10x+12+6x+1010x + 12 + 6x + 10Step 2: Combine like termsNow, group the like terms (terms involving xx and constant terms):•Combine the xx-terms: 10x+6x=16x10x + 6x = 16x•Combine the constants: 12+10=2212 + 10 = 22Step 3: Write the simplified resultSo, the simplified expression is:16x+2216x + 22Final Answer:16x+2216x + 222.Subtract the polynomialLet's break down the given expression step by step:Given: (11x2+7x+11)−(11x2+20x+12)(11x^2 + 7x + 11)-(11x^2 + 20x + 12)Step 1:Distribute the negative sign

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Distribute the negative sign across the second set of parentheses:11x2+7x+11−11x2−20x−1211x^2 + 7x + 11-11x^2-20x-12Step 2: Combine like termsNow, group the like terms (terms involving x2x^2, xx, andconstants):•Combine the x2x^2-terms: 11x2−11x2=011x^2-11x^2 = 0•Combine the xx-terms: 7x−20x=−13x7x-20x =-13x•Combine the constant terms: 11−12=−111-12 =-1Step 3: Write the simplified resultSo, the simplified expression is:−13x−1-13x-1Final Answer:−13x−1-13x-13.SimplifyLet's simplify 12×14412\times\sqrt{144}.Step 1: Find the square root of 144144=12\sqrt{144} = 12Step 2: Multiply by 12

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12×12=14412\times 12 = 144Final Answer:1441444.Find the distance between Pand Q using the distance formulaP(-3,8) and Q(5,8)To find the distance between the points P(−3,8)P(-3, 8) and Q(5,8)Q(5, 8) using the distance formula, we'llfollow the steps below:Distance Formula:The formula is:d=(x2−x1)2+(y2−y1)2d =\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}Where:•(x1,y1)(x_1, y_1) is the first point (P),•(x2,y2)(x_2, y_2) is the second point (Q).Step 1: Identify the coordinates of points P and Q•P(x1,y1)=(−3,8)P(x_1, y_1) = (-3, 8)•Q(x2,y2)=(5,8)Q(x_2, y_2) = (5, 8)Step 2: Plugthe coordinates into the distance formulaNow substitute the values of x1x_1, y1y_1, x2x_2, and y2y_2 into the formula:d=(5−(−3))2+(8−8)2d =\sqrt{(5-(-3))^2 + (8-8)^2}Step 3: Simplify the termsFirst, simplify inside the parentheses:

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