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Quadratic Equations and Algebraic Expressions, Exams of Algebra

Solutions to various algebraic expressions and quadratic equations, including performing operations, simplifying answers, finding quotients and remainders through long division, and solving equations. It covers topics such as polynomial expressions, quadratic equations, and algebraic manipulation.

Typology: Exams

2021/2022

Uploaded on 09/27/2022

agrata
agrata ๐Ÿ‡บ๐Ÿ‡ธ

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1. Perform the indicated operation and simplify your answer:
a. x2โˆ’6x+ 9
x2โˆ’3xโˆ’10 ยทx2โˆ’4xโˆ’5
x2โˆ’2xโˆ’3
Ans: xโˆ’3
x+ 2
b. x2+ 2xโˆ’15
x2+xโˆ’12 ยทx2โˆ’25
x+ 4
Ans: (xโˆ’5)(x+ 5)2
(x+ 4)2
c. 2โˆ’x
x2โˆ’2xยทx2+ 4x+ 4
x2โˆ’4
Ans: โˆ’(x+ 2)
x(xโˆ’2)
d. x2+ 3x+ 2
2x3รท(x+ 1)2
10x
Ans: 5(x+ 2)
x2(x+ 1)
e. 10 โˆ’20x
x2โˆ’2xรท2x2+ 7xโˆ’4
x2+ 2xโˆ’8
Ans: โˆ’10
x
f. 4x2+ 10
xโˆ’3รท6x2+ 15
x2โˆ’9
Ans: 2(x+ 3)
3
2. Find the quotient and remainder by performing long division:
a. (x2โˆ’3xโˆ’10) รท(x+ 1)
Ans: (xโˆ’4) โˆ’6
x+ 1
b. (โˆ’2x5+ 3x4โˆ’2x2+xโˆ’1) รท(x2โˆ’1)
Ans: (โˆ’2x3+ 3x2โˆ’2x+ 1) โˆ’x
x2โˆ’1
c. (3x6+ 1) รท(x2โˆ’2x+ 2)
Ans: (3x4+ 6x3+ 6x2โˆ’12) + โˆ’24x+ 25
x2โˆ’2x+ 2
d. (โˆ’x3+ 4x2+ 4x+ 4) รท(x2+x+ 1)
Ans: (โˆ’x+ 5) โˆ’1
x2+x+ 1
pf3

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  1. Perform the indicated operation and simplify your answer:

a. x^2 โˆ’ 6 x + 9 x^2 โˆ’ 3 x โˆ’ 10

x^2 โˆ’ 4 x โˆ’ 5 x^2 โˆ’ 2 x โˆ’ 3

Ans: x^ โˆ’^3 x + 2

b. x^2 + 2x โˆ’ 15 x^2 + x โˆ’ 12

x^2 โˆ’ 25 x + 4

Ans: (x โˆ’ 5)(x + 5)^2 (x + 4)^2

c. 2 โˆ’ x x^2 โˆ’ 2 x

x^2 + 4x + 4 x^2 โˆ’ 4

Ans: โˆ’(x + 2) x(x โˆ’ 2)

d. x^2 + 3x + 2 2 x^3

รท

(x + 1)^2 10 x

Ans: 5(x + 2) x^2 (x + 1)

e. 10 โˆ’ 20 x x^2 โˆ’ 2 x

รท

2 x^2 + 7x โˆ’ 4 x^2 + 2x โˆ’ 8

Ans: โˆ’^10 x

f. 4 x^2 + 10 x โˆ’ 3

รท

6 x^2 + 15 x^2 โˆ’ 9

Ans: 2(x + 3) 3

  1. Find the quotient and remainder by performing long division:

a. (x^2 โˆ’ 3 x โˆ’ 10) รท (x + 1)

Ans: (x โˆ’ 4) โˆ’

x + 1 b. (โˆ’ 2 x^5 + 3x^4 โˆ’ 2 x^2 + x โˆ’ 1) รท (x^2 โˆ’ 1)

Ans: (โˆ’ 2 x^3 + 3x^2 โˆ’ 2 x + 1) โˆ’ x x^2 โˆ’ 1 c. (3x^6 + 1) รท (x^2 โˆ’ 2 x + 2)

Ans: (3x^4 + 6x^3 + 6x^2 โˆ’ 12) + โˆ’ 24 x + 25 x^2 โˆ’ 2 x + 2 d. (โˆ’x^3 + 4x^2 + 4x + 4) รท (x^2 + x + 1)

Ans: (โˆ’x + 5) โˆ’

x^2 + x + 1

e. (โˆ’ 4 x^4 + 3x^3 โˆ’ x) รท (x โˆ’ 3)

Ans: (โˆ’ 4 x^3 โˆ’ 9 x^2 โˆ’ 27 x โˆ’ 82) โˆ’

x โˆ’ 3

  1. Perform the indicated operation and simplify your answer:

a.

x^2 โˆ’ 5 x + 6 โˆ’^

x โˆ’ 2

Ans: 10 โˆ’ 3 x (x โˆ’ 2)(x โˆ’ 3)

b.

x โˆ’ 3

x + 1 3 โˆ’ x

Ans: x^ + 5 x โˆ’ 3

c. 2 x x + 2 +^

x + 2 Ans: 2

d. 3 6 x^2 โˆ’ 4 x โˆ’ x^ โˆ’^2 9 x โˆ’ 6

Ans: โˆ’ 2 x^2 + 4x + 9 6 x(3x โˆ’ 2)

e. x^2 x โˆ’ 3

x^2 3 โˆ’ x Ans: 0

  1. Solve the equation. Be sure to check for extraneous solution:

a.^7 x

  • x = โˆ’ 8

Ans: {โˆ’ 7 , โˆ’ 1 }

b. 1 x โˆ’ 1

+^2

x = x x โˆ’ 1 Ans: { 2 }

c.

x โˆ’ 2 +

x = 2 Ans: { 1 , 4 }

d.

x โˆ’ 2

x + 3

x^2 + x โˆ’ 6 Ans: {โˆ’ 4 }

e.

2 x โˆ’ 4

2 โˆ’ x