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Simplifying Polynomials and Solving Quadratics: Comprehensive Guide

This guide provides step-by-step methods for simplifying polynomials, writing them in standard form, and finding sums or differences of algebraic expressions. It covers various approaches, including factoring, completing the square, and using the quadratic formula. Key examples include polynomial addition and subtraction, factoring techniques, scientific notation conversion, and real-world problem solving such as calculating sales prices and measuring distances. Suitable for students and enthusiasts looking to strengthen their algebra skills.

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Simplifying Polynomials and Solving Quadratics: Comprehensive Guide

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  1. Simplify the polynomial and write it in standard form. −3a − 2 + a2− 4a + 6 + 5a2 1 = 6a2 −7a + 4 Find the sum or difference. A. (3x2 − 8x + 2) + ( x2+ 5x − 3) 1 = 4x2 − 3x − 1 B. (5x − 6y) + (2x + 8y) = 7x + 2y C. = −6x2 + 14 (2x2 + 5) −(8x2 − 9) +− = (2x2 + 5) + (−8x2 + 9) (9n2− 3n) − (3n2 − 5n + 4) +− = 6n2 + 2n − 4 D. (9n2− 3n) + (−3n2+ 5n − 4)

  2. Simplify: B. −4a2 ( 2a2 ─ 3a) +− A.(3x2y4)(2x3y5) (−4a2) ∙ 2a2 + (−4a2)∙ (−3a ) 1 3 ∙ 2 ∙ x2+3 ∙ y4+5 = −8a4 + 12a3 = 6 x5y9 D. (x + 3)2 C. (x + 3)(x + 3) + 3x + 3x + 9 x2 x2 +6x + 9 D. (2y − 2)(y2 − 3y + 4) 2y − 2 2y∙ y2 + 2y∙(−3y) + 2y∙4 + (−2) ∙ y2 + (−2)(−3y) + (−2) ∙ 4 2y3 + (−6y2) + 8y + (−2y2) + 6y + (−8) 2y3 − 8y2 + 14y − 8

  3. 6a3 − 15a2 + 9a −3a 6a3 −15a2 9a = + + −3a −3a −3a = −2a2 + 5a + − 3 − Simplify: Write the answers using positive exponents. B. −22x−3∙ x0 C. A. (−2x3y2)2 (−2)2∙x3∙2∙y2∙2 = −5a5−2∙b2−3 =−5a3b−1 D.

  4. Simplify: Write the answers using positive exponents. C. A. B. D. Write 6,031,000 in Scientific Notation: 6.031 x 106 E. Change 4.2 x 10−3 into decimal form: 0.0042

  5. A. Give the greatest common factor of the monomials: B. Factor Completely: 18x3y2, 30x2y5 27a3b2− 12ab2 3ab2(9a2− 4) GCF: 6x2y2 3ab2(3a + 2)(3a − 2) C. Factor: D. Factor: E. Factor: x2 − 25 (x + 5)(x −5) (x + 1)(x −4) (x + 2)(x + 6) F. Factor Completely: 2x3 +6x2 − 8x 2x(x2 +3x− 4) 2x(x+4)(x− 1)

  6. Solve: Solve: A. x2 + 5x − 24 = 0 B. 2x2 − 2x − 12 = 0 (x − 3)(x + 8) = 0 2(x2− x − 6) = 0 x − 3 = 0 or x + 8 = 0 2(x + 2)(x −3) = 0 x = 3 or x = −8 x + 2 = 0 or x − 3 = 0 Sol. Set: {−8, 3} x = −2 or x = 3 Sol. Set: {−2, 3} Solve by completing the square: C. x2 − 6x = 1

  7. Simplify: B. A. Solve: ] [ D. 6 ∙ C. 10x ∙ 20 − 2x = 4 −2x = −16 x = 8

  8. Simplify: C. B. A. Solve: E. D. ] [ ∙ (x+5)(x−5) 2∙(x+5) = 4 2 x + 10 = 4 2 x = −6 x = −3

  9. A. What is the sales price of a $40 sweater that is on sale for 33% off? 33% of $40 = $40 − $13.20 = $26.80 .33 ∙ $40 = $13.20 $26.80 sales price B. 30 is 125% of what number. 30 = 1.25 ∙ n n = 24 C. The height of a triangle is 3 cm more than its base. If the area of the triangle is 90 cm2, write an equation that can be used to find the measure of the base, b and solve for b. b = 12 cm

  10. 13 ft ? √ √ 5 ft The top of a 13 ft ladder is propped against the side of a house. The bottom of the ladder is 5 feet from the house. How far up the wall does the top of the ladder touch the house? a2 + b2 = c2 52 + b2 = 132 c= = b 25 + b2 = 169 a= b2 = 144 b = 12 ft

  11. Simplify: D. C. B. A. F. E. G. Rationalize and simplify:

  12. √ √ Solve: C. (x + 4)2 = 16 x + 4 = ±4 x − 1 = 100 x + 4 = 4 or x + 4 = −4 x = 101 x = 0 or x = −8 D. 6n2 − 150 = 0 6n2 = 150 n2 = 25 n = 5, −5 3x + 1 = 100 3x = 99 x = 33

  13. x1 + x2 4 + 2 5+ 7 y1 + y2 ( ) ( ) 2 2 2 2 , , x A. Find the distance between the points A(1, 1) and B(3, 4) x2 = 22 + 32 x 3 2 B. Find the midpoint of the segment with the following endpoints: (4, 5) and (2, 7). midpoint: = (3, 6)

  14. Solve using the quadratic formula: ax2 +bx + c = 0 x2 −3x −7 = 0 a = 1, b = −3, and c = −7

  15. Use the discriminant to find the number of real number roots of this equation: ax2 +bx + c = 0 Discriminant: b2 − 4ac 5x2 +10x + 5 = 0 = 102 − 4∙5∙5 a = 5, b = 10, and c = 5 = 100 − 100 = 0 The discriminant is equal to “0” so there is one real number root.

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