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ISEE Upper Level Math : Algebraic Concepts

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Example Questions

Example Question #291 : Algebraic Concepts

Exponentiate:

$\left (x + 0.5y \right) ^{3}$

Possible Answers:

$x^{3} + 0.125 y ^{3}$

$x^{3} + 0.5 x^{2} y + 0.25 \ x y ^{2} + 0.125 y ^{3}$

$x^{3} + x^{2} y + 0.5 \ x y ^{2} + 0.125 y ^{3}$

$x^{3} + 1.5 x^{2} y + 0.75 \ x y ^{2} + 0.125 y ^{3}$

$x^{3} + 1.5 x^{2} y + 0.75 \ x y ^{2} + 1.25 y ^{3}$

Correct answer:

$x^{3} + 1.5 x^{2} y + 0.75 \ x y ^{2} + 0.125 y ^{3}$

Explanation:

The cube of a sum pattern can be applied here:

$\left (x + 0.5y \right) ^{3}$

$= x^{3} + 3 \cdot x^{2}\cdot 0.5y + 3 \cdot x \cdot\left (0.5y \right) ^{2} + \left (0.5y \right) ^{3}$

$= x^{3} + 3\cdot 0.5 \cdot x^{2} \cdot y + 3 \cdot 0.5^{2} \cdot x \cdot y ^{2} + 0.5^{3} \cdot y ^{3}$

$= x^{3} + 3 \cdot 0.5 x^{2} y + 3 \cdot 0.25 \ x y ^{2} + 0.125 y ^{3}$

$= x^{3} + 1.5 x^{2} y + 0.75 \ x y ^{2} + 0.125 y ^{3}$

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Example Question #292 : Algebraic Concepts

Write in expanded form:

$\left (2x + \frac{1}{2} \right )^{3}$

Possible Answers:

$8x^{3} + 6 x^{2} + \frac{3}{2} x + \frac{1}{8}$

$8x^{3} + 2 x^{2} + \frac{1}{2} x + \frac{1}{8}$

$8x^{3} + \frac{1}{8}$

$8x^{3} + x^{2} +x + \frac{1}{8}$

$8x^{3} + 6 x^{2} + 3x + \frac{1}{8}$

Correct answer:

$8x^{3} + 6 x^{2} + \frac{3}{2} x + \frac{1}{8}$

Explanation:

The cube of a sum pattern can be applied here:

$\left (2x + \frac{1}{2} \right )^{3}$

$= \left ( 2x\right ) ^{3} + 3\cdot \left ( 2x\right ) ^{2} \cdot \left ( \frac{1}{2} \right ) + 3\cdot \left ( 2x\right ) \cdot \left ( \frac{1}{2} \right ) ^{2} + \left ( \frac{1}{2} \right ) ^{3}$

$= 2^{3}\cdot x^{3} + 3\cdot 2^{2}\cdot \frac{1}{2} \cdot x^{2} + 3\cdot2 \cdot \left (\frac{1}{2} \right ) ^{2} \cdot x \left + \left ( \frac{1}{2} \right ) ^{3}$

$= 8x^{3} + 6 x^{2} + \frac{3}{2} x + \frac{1}{8}$

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Example Question #293 : Algebraic Concepts

Factor completely:

$x ^{2} - 11x + 20$

Possible Answers:

(x - 4)(x - 5)

(x + 2)(x - 10)

(x - 4)(x + 5)

(x - 2)(x - 10)

The expression cannot be factored.

Correct answer:

The expression cannot be factored.

Explanation:

A trinomial with leading coefficient $x^{2}$ can be factored by looking for two integers to fill in the boxes:

$(x + \square )(x + \square )$ .

The numbers should have product 20 and sum . However, all of the possible factor pairs fail:

-1 + (-20) = -21

-2 + (-10) = -12

-4 + (-5) = -9

The polynomial is prime.

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Example Question #3 : How To Multiply Exponential Variables

Factor completely:

$4x^{2} - 36x+ 81$

Possible Answers:

(2x-3)(2x-27)

(4x-9)(x-9)

(4x-3)(x-27)

$(2x-9)^{2}$

(x-3)(4x-27)

Correct answer:

$(2x-9)^{2}$

Explanation:

$4x^{2} - 36x+ 81$ can be seen to be a perfect square trinomial by taking the square root of the first and last terms, multiplying their product by 2, then comparing it to the second term:

$4x^{2} = \left ( 2x\right )^{2}$

$81 = 9^{2}$

$2 \cdot2x\cdot9= 36x$

Therefore,

$4x^{2} - 36x+ 81$

$=\left ( 2x \right ) ^{2} -2 \cdot 2 x \cdot9+ 9 ^{2}$

$=(2x-9)^{2}$

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Example Question #11 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

$x ^{2} + 20x + \square$

Possible Answers:

200

100

Correct answer:

100

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 20, by 2, and square the quotient. The result is

$\left (\frac{20}{2} \right )^{2} =10 ^{2} =100$

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Example Question #12 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

$x ^{2} + 9x + \square$

Possible Answers:

$\frac{81}{2}$

$\frac{81}{4}$

Correct answer:

$\frac{81}{4}$

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is 9, by 2, and square the quotient. The result is

$\left (\frac{9}{2} \right) ^{2} = \frac{9^{2}}{2^{2}}= \frac{81}{4}$

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Example Question #13 : How To Multiply Exponential Variables

Multiply:

(x + y + z) (x + y)

Possible Answers:

$x^{2}+ xy+ 2yz + xz + y^{2}$

$x^{2}+3 xyz + y^{2}$

$x^{2}+ xy+ yz + 2xz + y^{2}$

$x^{2}+2 xy+ yz + xz + y^{2}$

$x^{2} + xy + yz + xz + y^{2}$

Correct answer:

$x^{2}+2 xy+ yz + xz + y^{2}$

Explanation:

Use the distributive property, then collect like terms:

(x + y + z) (x + y)

$= (x + y + z) \cdot x + (x + y + z) \cdot y$

$= x\cdot x + y \cdot x + z\cdot x + x \cdot y+ y \cdot y+ z \cdot y$

$= x^{2}+ xy+ xz + x y+ y^{2}+ yz$

$= x^{2}+ xy+ x y + yz + xz + y^{2}$

$= x^{2}+2 xy+ yz + xz + y^{2}$

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Example Question #14 : How To Multiply Exponential Variables

Multiply:
(6x^3)(x^3)^2

Possible Answers:

$6x^{12}$

6x^8

6x^9

$6x^{18}$

Correct answer:

6x^9

Explanation:

(6x^3)(x^3)^2=(6x^3)(x^6)=6x^9

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Example Question #294 : Algebraic Concepts

Simplify:

$(0.5x-0.4)^{2}$

Possible Answers:

$2.5x^{2} + 0.4x- 1.6$

$0.25x^{2}+ 0.4x- 0.16$

$2.5x^{2} - 0.4x+ 1.6$

$0.25x^{2} - 0.4x+ 0.16$

$0.25x^{2} - 0.16$

Correct answer:

$0.25x^{2} - 0.4x+ 0.16$

Explanation:

Use the pattern, substituting A = 0.5x,B = 0.4 .

$(A - B)^{2} = A^{2} - 2AB + B^{2}$

$(0.5x - 0.4)^{2} = 0.5x^{2} - 2\cdot 0.5x \cdot 0.4+ 0.4^{2}$

$= 0.25x^{2} - 0.4x+ 0.16$

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Example Question #15 : How To Multiply Exponential Variables

Write in expanded form.

$(x\neq 0)$

$(4x+\frac{3}{x})^3$

Possible Answers:

$64x^3+144x+\frac{108}{x}-\frac{27}{x^3}$

$64x^3-144x+\frac{108}{x}-\frac{27}{x^3}$

$64x^3+144x+\frac{108}{x}+\frac{27}{x^2}$

$64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

$64x^3+172x+\frac{108}{x}+\frac{27}{x^3}$

Correct answer:

$64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

Explanation:

$(4x+\frac{3}{x})^3$

$=(4x)^3+3(4x)^2(\frac{3}{x})+3(4x)(\frac{3}{x})^2+(\frac{3}{x})^3$

$=4^3\times x^3+3\times 4^2\times 3\times \frac{x^2}{x}+3\times 4\times 3^2\times \frac{x}{x^2}+\frac{3^3}{x^3}$

$=64x^3+144x+\frac{108}{x}+\frac{27}{x^3}$

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ISEE Upper Level Math : Algebraic Concepts

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #291 : Algebraic Concepts

Example Question #292 : Algebraic Concepts

Example Question #293 : Algebraic Concepts

Example Question #3 : How To Multiply Exponential Variables

Example Question #11 : How To Multiply Exponential Variables

Example Question #12 : How To Multiply Exponential Variables

Example Question #13 : How To Multiply Exponential Variables

Example Question #14 : How To Multiply Exponential Variables

Example Question #294 : Algebraic Concepts

Example Question #15 : How To Multiply Exponential Variables

All ISEE Upper Level Math Resources

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