GRE - Basic Algebra Refresher
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GRE
Basic Algebra Refresher
Simplifying expressions
• “Like terms” are terms that have the same variables
Examples:
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4x and 7x are like terms because their variable “x ” are identical |
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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
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Practice: Simplify the following |
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3x + 2x |
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5x |
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b) 5x −2y − x + 9y |
b) 4x + 7y |
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c) −3m + 6 −2 + m |
c) −2m + 4 |
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d) |
1 x + |
3 x |
d) 4 x |
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5 |
5 |
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5 |
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e) |
5xy2 −8xy2 + xy2 |
e)−2xy2 |
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f) |
x2y3 −2x5 y7 +6y3 x2 + 5y7 x5 |
f) |
7x2y3 +3x5 y7 |
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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)
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Examples: |
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(3x2 )(5x7 )= 15x9 |
Don’t forget that if a variable has no |
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(m5 )( |
4m7 )( |
6m3 )= 24m15 |
exponent shown, then the exponent is 1 |
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e.g., x = x 1 |
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(8n)(n10 )= 8n11 |
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(7x2 y 5z3 )(2x4 y6z)= 14x6y11z4 |
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Practice: Simplify the following |
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a) (5y3 )(8y6 ) |
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a) 40y9 |
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b) (4 j2k3m5 )(9 j6k4m) |
b) 36 j8k7m6 |
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c) (−4x3y3 )(x2y5 )(5xy3 ) |
c) −20x6y11 |
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d) (x2y3 )(2x5 y7 )(5y7 x5 ) |
d) 10x12y17 |
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e) |
(4x2y )(3x )(2y5z2 ) |
e) 24x3y6z2 |
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Practice: Fill in the blanks |
Answers: |
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a) (9y5 )( |
)= 18y8 |
a) (9y5 )(2y3 )= 18y8 |
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b) |
(4x5 )(2x )( |
) = 40x12 |
b) (4x5 )(2x )(5x6 )= 40x12 |
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c) ( |
)(6x3y2 )= −24x8y3 |
c) (−4x5 y )(6x3y2 )= −24x8y3 |
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d) |
(−10xyz2 )( |
)= 30x3y5z2 |
d) (−10xyz2 )(−3x2y 4 )= 30x3y5z2 |
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e) |
(5yz)( |
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) = 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 |
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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” |
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−7x (x2 + 2x −5)= −7x3 −14x2 + 35x |
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4x3y2z (5xyz − x3z2 )= 20x4 y 3z2 − 4x6y 2z3 |
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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
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a) −2x ( |
) = −4x6 +6x |
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b) 4x3 ( |
)= 20x8 +16x5 −12x3 |
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c) 5x3y ( |
) = −10x7y +5x3y2 |
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d) 4xz ( |
) = −4x4yz + 8xyz4 |
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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 |
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Basic Algebra Refresher |
Multiplying two binomials |
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• One by one, multiply a term from one binomial with a term from the other binomial |
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• Multiply terms in the following order Front, Outside, Inside, Last (F.O.I.L.) |
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(x +3)(x +7) |
(x +3)(x +7) |
(x +3)(x +7) |
(x +3)(x +7) |
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First |
Outside |
Inside |
Last |
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x2 |
7x |
3x |
21 |
Examples: |
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(x + 3)(x +7) = x2 +7x + 3x +21 |
(x −2)(x + 9)= x2 +9x −2x −18 |
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= x2 +10x + 21 |
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= x2 +7x −18 |
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(3x −5)(2x +1) = 6x2 + 3x −10x −5 |
(5x −2)(3x −7) = 15x2 −35x −6x +14 |
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= 6x2 −7x −5 |
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= 15x2 − 41x +14 |
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Practice: Expand and simplify the following |
Answers: |
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a) (x + 6)(x + 2) |
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a) x2 + 8x +12 |
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b) (2x +1)(x +5) |
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b) 2x2 +11x + 5 |
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c) (x − 4)(x +3) |
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c) x2 − x −12 |
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d) (3x −5)(2x −1) |
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d) 6x2 −13x + 5 |
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e) (3x +7)(x +1) |
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e) 3x2 +10x + 7 |
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f) |
(5x +2)(5x −2) |
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f) 25x2 − 4 |
g) |
(4x2 −3)(2x +5) |
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g) 8x3 + 20x2 −6x −15 |
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