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ISEE Upper Level Math : How to divide variables

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Example Question #1 : How To Divide Variables

$\dpi{100} \frac{1}{3}\div \frac{1}{9}$

Possible Answers:

$\dpi{100} 2$

$\dpi{100} 3$

$\dpi{100} \frac{1}{27}$

$\dpi{100} \frac{1}{3}$

Correct answer:

$\dpi{100} 3$

Explanation:

Never divide fractions! Simply flip the fraction that follows the division symbol, and then multiply it by the first fraction. So this expression becomes:

$\dpi{100} \frac{1}{3}\times \frac{9}{1}=\frac{9}{3}=3$

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Example Question #2 : How To Divide Variables

Divide:

$\frac{6x^3-3x^2+9x}{3x}$

Possible Answers:

2x^3-x^2+3

2x^3+x^2+3

2x^2+x+3

2x^2-x+3

Correct answer:

2x^2-x+3

Explanation:

$\frac{6x^3-3x^2+9x}{3x}=\frac{6x^3}{3x}-\frac{-3x^2}{3x}+\frac{9x}{3}=2x^2-x+3$

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Example Question #3 : How To Divide Variables

Simplify:

$\frac{4x^{4}- 8 x^{2} - 16} {2x}$

Possible Answers:

$2x^{3} - 4 x - \frac{8} {x}$

$2x^{3} - 4 x^{2} - \frac{8} {x}$

$2x^{3} - 12x$

$2x^{3} - 4 x^{2} -8$

$2x^{3} - 4 x -8$

Correct answer:

$2x^{3} - 4 x - \frac{8} {x}$

Explanation:

$\frac{4x^{4}- 8 x^{2} - 16} {2x}$

$= \frac{4x^{4}} {2x} - \frac{ 8 x^{2}} {2x} - \frac{16} {2x}$

$= \frac{2x^{4-1}} {1} - \frac{ 4 x^{2-1}} {1} - \frac{8} {x}$

$= \frac{2x^{3}} {1} - \frac{ 4 x^{1}} {1} - \frac{8} {x}$

$= 2x^{3} - 4 x - \frac{8} {x}$

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Example Question #1 : How To Divide Variables

If $x\neq 0$ , divide:

$\frac{8x^4+4x^3-16x}{4x}$

Possible Answers:

x^3+x^2-4

2x^3-x^2-4

2x^3+x^2+4

2x^3+x^2-4

2x^3-x^2+4

Correct answer:

2x^3+x^2-4

Explanation:

$\frac{8x^4+4x^3-16x}{4x}$

$=\frac{8x^4}{4x}+\frac{4x^3}{4x}-\frac{16x}{4x}$

=2x^3+x^2-4

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Example Question #5 : How To Divide Variables

Divide:

$\frac{16y^3 - 4y^2 + 8y}{4y}$

Possible Answers:

12y^2 - y + 4

y^2 - 2y + 8

4y^3 - y^2 + 2y

4y^2 + 2

4y^2 - y + 2

Correct answer:

4y^2 - y + 2

Explanation:

In this division problem, you can simplify first the coefficients, then the variables.

Each of the coefficients in the numerator is divisible by the coefficient in the denominator, allowing you to divide out and cancel the 4 in the denominator:

$\frac{16y^3 - 4y^2 + 8y}{4y} = \frac{4y^3 - y^2 + 2y}{y}$

Finally, you can simplify the varaibles. Remember that when simplifying variables in a fraction (division problem), you subtract the numerator variable's exponent by the denominator variable's exponent. You can do this with each term in this problem, because each term in the numerator has at least a :

$\frac{4y^3 - y^2 + 2y}{y} = 4y^2 - y +2$

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Example Question #6 : How To Divide Variables

Divide:

$\frac{7x^4 + 21x^2 - 14}{7x^2}$

Possible Answers:

3x^2 + 3

x^2 + 1

$x^2 + 3 - \frac{2}{x^2}$

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

Correct answer:

$x^2 + 3 - \frac{2}{x^2}$

Explanation:

In this division problem, you can simplify first the coefficients, then the variables.

Each of the coefficients in the numerator is divisible by the coefficient in the denominator, allowing you to divide out and cancel the 7 in the denominator:

$\frac{7x^4 + 21x^2 - 14}{7x^2} = \frac{x^4 + 3x^2 - 2}{x^2}$

You can only do this with the first two terms in the numerator, since the final term does not have a variable. Instead, the final term will keep x^2 in the denominator:

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

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Example Question #3 : How To Divide Variables

Divide:

$\frac{x^2+2xy+xy^2}{x}$

Possible Answers:

x+2y+y^2

2x+2y+y^2

x^2+2y+y^2

2x^2+2y+y^2

Correct answer:

x+2y+y^2

Explanation:

$\frac{x^2+2xy+xy^2}{x}=\frac{x^2}{x}+\frac{2xy}{x}+\frac{xy^2}{x}=x+2y+y^2$

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Example Question #24 : Variables

Which of the following is a factor of $(8-2)^{2}$ ?

Possible Answers:

Correct answer:

Explanation:

The first step to solving this question is to reduce $(8-2)^{2}$ .

$(8-2)^{2}$

$6^{2}$

The only number listed that is a factor of 36 is 18, given that 2 times 18 is 36. Therefore, 18 is the correct answer.

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Example Question #31 : Operations

Divide:

$\frac{4x^{3}+8x^{2}-12x}{4x}$

Possible Answers:

$4x^{2}+8x-12$

$x^{2}+2x-3$

$x^{3}+2x^{2}-3x$

The expression cannot be simplified further.

$x^{4}+2x^{3}-3x^{2}$

Correct answer:

$x^{2}+2x-3$

Explanation:

To divide this problem we simplify it first. In this problem we can separate the big fraction into three smaller fractions.

$\frac{4x^{3}+8x^{2}-12x}{4x}=\frac{4x^{3}}{4x}+\frac{8x^{2}}{4x}-\frac{12x}{4x}$

Then from here, we can pull out a from both the numerator and denominator of each smaller fraction.

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

Now we cancel terms and get the following result:

x^2+2x-3

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Example Question #10 : How To Divide Variables

Simplify:

$\frac{2x^{4}+6x^{3}+8}{2x}$

Possible Answers:

$x^{3}+3x^{2}+4x$

$4x^{3}+12x^{2}+16x$

$4x^{3}+6x^{2}+16x$

$x^{3}+3x^{2}+4$

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

Correct answer:

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

Explanation:

To simplify this problem we first separate the large fraction into three smaller fractions.

$\frac{2x^{4}+6x^{3}+8}{2x}$

$=\frac{2x^{4}}{2x}+\frac{6x^{3}}{2x}+\frac{8}{2x}$

From here we can factor out from the numerator and denominator of the first two fractions.

$= \frac{2x(x^3)}{2x}+\frac{2x(3x^2)}{2x}+\frac{8}{2x}$

The can be canceled out in the first two fractions. From here we can factor out a from the numerator and denominator of the third fraction.

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

Thus becoming:

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

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ISEE Upper Level Math : How to divide variables

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Divide Variables

Example Question #2 : How To Divide Variables

Example Question #3 : How To Divide Variables

Example Question #1 : How To Divide Variables

Example Question #5 : How To Divide Variables

Example Question #6 : How To Divide Variables

Example Question #3 : How To Divide Variables

Example Question #24 : Variables

Example Question #31 : Operations

Example Question #10 : How To Divide Variables

All ISEE Upper Level Math Resources

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