GRE - Basic Algebra Refresher

GRE

Basic Algebra Refresher

Simplifying expressions

• “Like terms” are terms that have the same variables

Examples:

●	4x and 7x are like terms because their variable “x ” are identical
●	9xy and 2xy are like terms because their variables “xy ” are identical

●x 2y 5, 7x 2y 5 and –13x 2y 5 are like terms because their variables “x 2y 5” are identical

•When there is no number in front of the variable (called the coefficient), that number is assumed to be 1

Examples:

●x = 1x

●x 2y 5 = 1x 2y 5

•Like terms can be combined, by adding or subtracting coefficients. This process is called “simplifying.”

● 4x + 7x = 11x

●9xy – 2xy = 7xy

●x 2y 5+ 7x 2y 5 –13x 2y 5 = –5x 2y 5

	Practice: Simplify the following			Answers:
	a)	3x + 2x		a)	5x
	b) 5x −2y − x + 9y			b) 4x + 7y
	c) −3m + 6 −2 + m			c) −2m + 4
	d)	1 x +	3 x	d) 4 x
		5	5		5
	e)	5xy2 −8xy2 + xy2		e)−2xy2
	f)	x2y3 −2x5 y7 +6y3 x2 + 5y7 x5		f)	7x2y3 +3x5 y7
	g) x + 6y + 7x −5y −8x			g) y

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GRE

Basic Algebra Refresher

Multiplying monomials (multiplying one term by another)

• Multiply the coefficients and multiply similar variables (remember your exponent laws)

	Examples:
	(3x2 )(5x7 )= 15x9					Don’t forget that if a variable has no
	(m5 )(		4m7 )(	6m3 )= 24m15		exponent shown, then the exponent is 1
						e.g., x = x 1
	(8n)(n10 )= 8n11
	(7x2 y 5z3 )(2x4 y6z)= 14x6y11z4
	Practice: Simplify the following					Answers:
	a) (5y3 )(8y6 )					a) 40y9
	b) (4 j2k3m5 )(9 j6k4m)					b) 36 j8k7m6
	c) (−4x3y3 )(x2y5 )(5xy3 )					c) −20x6y11
	d) (x2y3 )(2x5 y7 )(5y7 x5 )					d) 10x12y17
	e)	(4x2y )(3x )(2y5z2 )				e) 24x3y6z2
	Practice: Fill in the blanks					Answers:
	a) (9y5 )(			)= 18y8		a) (9y5 )(2y3 )= 18y8
	b)	(4x5 )(2x )(			) = 40x12	b) (4x5 )(2x )(5x6 )= 40x12
	c) (		)(6x3y2 )= −24x8y3			c) (−4x5 y )(6x3y2 )= −24x8y3
	d)	(−10xyz2 )(			)= 30x3y5z2	d) (−10xyz2 )(−3x2y 4 )= 30x3y5z2
	e)	(5yz)(			) = 5wx2y3z2	e) (5yz)(wx2y2z)= 5wx2y3z2

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GRE

Basic Algebra Refresher

Multiplying a polynomial by a monomial

• Multiply each term in the polynomial by the monomial in front.

Examples:

5(2x + 7) = 10x + 35
2x3 (7x2 −3x )= 14x5 −6x4	The process of multiplying each
−10(m6 −2m5 )= −10m6 + 20m5	term in the parentheses by the
	term in front is called “expanding”
−7x (x2 + 2x −5)= −7x3 −14x2 + 35x
4x3y2z (5xyz − x3z2 )= 20x4 y 3z2 − 4x6y 2z3

Practice: Simplify the following

a)3(2y2 + 4y −7)

b)2x (x3 −5x )

c)−5x2y6 (3xy − x7y2 − x2 )

d)−6xy2 (−3 + 2x − y7 )

e)7z2 (xz + 2x − 4wyz)

Practice: Fill in the blanks

	a) −2x (	) = −4x6 +6x
	b) 4x3 (	)= 20x8 +16x5 −12x3
	c) 5x3y (	) = −10x7y +5x3y2
	d) 4xz (	) = −4x4yz + 8xyz4

Answers:

a)6y2 +12y −21

b)2x4 −10x2

c)−15x3y7 + 5x9y8 + 5x4 y6

d)18xy2 −12x2y2 +6xy9

e)7xz3 +14xz2 −28wyz3

Answers:

a)−2x (2x5 −3)= −4x6 + 6x

b)4x3 (5x5 + 4x2 −3)= 20x8 +16x5 −12x3

c)5x3y (−2x4 + y )= −10x7y + 5x3y2

d)4xz (−x3y + 2yz3 )= −4x4yz + 8xyz4

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GRE				Basic Algebra Refresher
Multiplying two binomials
• One by one, multiply a term from one binomial with a term from the other binomial
• Multiply terms in the following order Front, Outside, Inside, Last (F.O.I.L.)
(x +3)(x +7)		(x +3)(x +7)	(x +3)(x +7)	(x +3)(x +7)
First		Outside	Inside	Last
	x2	7x	3x	21
Examples:
	(x + 3)(x +7) = x2 +7x + 3x +21		(x −2)(x + 9)= x2 +9x −2x −18
	= x2 +10x + 21			= x2 +7x −18
(3x −5)(2x +1) = 6x2 + 3x −10x −5			(5x −2)(3x −7) = 15x2 −35x −6x +14
	= 6x2 −7x −5			= 15x2 − 41x +14
Practice: Expand and simplify the following			Answers:
a) (x + 6)(x + 2)				a) x2 + 8x +12
b) (2x +1)(x +5)				b) 2x2 +11x + 5
c) (x − 4)(x +3)				c) x2 − x −12
d) (3x −5)(2x −1)				d) 6x2 −13x + 5
e) (3x +7)(x +1)				e) 3x2 +10x + 7
f)	(5x +2)(5x −2)			f) 25x2 − 4
g)	(4x2 −3)(2x +5)			g) 8x3 + 20x2 −6x −15
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