Simplifying Algebraic Expressions - GMAT Quantitative

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Question

Factor $\frac{x^{2}+6x+5}{x^{2}+10x+25}.$

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Answer

Let's first look at the numerator and denominator separately.

$x^{2}+6x+5$ : We need two numbers that multiply to 5 and add to 6. The numbers 1 and 5 work. So, $x^{2}+6x+5 = (x+5)(x+1)$

$x^{2}+10x + 25$ : We need two numbers that multiply to 25 and add to 10. The numbers 5 and 5 work. So, $x^{2}+10x + 25 = (x+5)(x+5)$

Putting this together, $\frac{x^{2}+6x+5}{x^{2}+10x+25} = \frac{(x+5)(x+1)}{(x+5)(x+5)} = \frac{x+1}{x+5}$

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Question

Find the solutions to the equation .

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Answer

Let's combine like terms.

, so the equation has no solution.

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Question

What is the simplified result of following the steps?

(1) Add to .

(2) Multiply the result by .

(3) Subtract $\left ( 2x+y \right )$ from the result.

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Answer

From (1), we can easily get the result .

Then from (2), we need to multiply by . This gives us .

The last step is to subtract $\left ( 2x+y \right )$ from :

$\left ( 16x+8y \right )-\left ( 2x+y \right )=16x+8y-2x-y=14x+7y$

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Question

$P = \frac{27D}{2d - D}$

Solve for .

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Answer

You have to isolate by moving around the separate components in the problem. The steps should go as follows:

$P = \frac{27D}{2d - D}$

$\frac{2dP}{27 + P} = D$

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Question

Let and be unknown variables. Simplify the following expression:

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Answer

To simplify algebraically, we combine like terms. First, we should get the expression in one long string, by removing the parentheses. So remembering the communitive property, the first group in parentheses will have no changes when we remove the parentheses. So simplifies to

However, note the second group in parentheses is being subtracted. So we must invert all the signs in the group to simplify properly. So the previous expression simplifies to

Finally we reorder and combine like terms to get

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Question

If you were to write $(x-3)^{10}$ in expanded form in descending order of degree, what would the third term be?

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Answer

By the Binomial Theorem, if you expand $(A+B)^{n}$ , writing the result in standard form, the $r \mathrm{th}$ term (with the terms being numbered from 0 to ) is

$C(n,r)A^{n-r}B^{r}$

Set , , , and (again, the terms are numbered 0 through , so the third term is numbered 2) to get

$C(10,2)\cdot x^{10-2}\cdot (-3)^{2}$

$=\frac{10!}{8! 2!}\cdot x^{8}\cdot 9$

$=45\cdot x^{8}\cdot 9$

$=405x^{8}$

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Question

If $x=\sqrt{\frac{25}{100}}$ ,

what is the value of

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Answer

Simplify.

$x=\sqrt{\frac{25}{100}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

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Question

Simplify

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Answer

Foil

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Question

Which answer is equivalent to $x^{40}$ ?

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Answer

$(x^a)^b=x^{ab}$

Therefore:

$(x^4)^{10}=x^{40}$

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Question

The sum of three consecutive integers is 12. What is the value of the middle integer?

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Answer

Let the value of the first integer be . This means that the consecutive integers will be , , and . The sum must be 12 which means that

$N+(N+1)+(N+2)=12\Rightarrow 3N+3=12\Rightarrow N=3$

Since the consecutive integers are 3, 4, and 5. The middle integer is 4.

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Question

Solve for .

$y=\frac{3x}{2+x}$

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Answer

$y=\frac{3x}{2+x}\Rightarrow y(2+x)=3x$

$2y+xy=3x\Rightarrow xy-3x=-2y$

$x(y-3)=-2y\Rightarrow x=\frac{-2y}{y-3}$

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Question

A number is divided by 4; its decimal point is then moved to the right 3 places. This is the same as doing what to the number?

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Answer

The best way to illustrate the answer to this question is to do these operations to the number 1.

First, divide by 4:

$1 \div 4 = 0.250$

Now move the decimal point right three spaces:

$0.250\Rightarrow 250.$

This has the effect of multiplying the number by 250.

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Question

Which of these expressions is equal to $\log_{7} x^{4}$ ?

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Answer

$\log_{7} x^{4} = 4 \log_{7} x = 4 \cdot\frac{ \ln x }{\ln 7} =\frac{ 4\ln x }{\ln 7}$

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Question

Assume that $x,y \neq 0$ .

Which of the following expressions is equal to the following expression?

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

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Answer

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

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

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

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

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

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

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Question

What is the coefficient of $x ^{5}$ in the expansion of $\left (x + 3 \right )^{8}$ ?

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Answer

By the Binomial Theorem, the $x^{5}$ term of $\left ( x+3\right ) ^{8}$ is:

$C(8,5) \cdot x^{5} \cdot 3 ^{8-5}$

The coefficient of $x ^{5}$ is therefore:

$C(8,5) \cdot 3 ^{8-5} = \frac{8!}{(8-5)!; 5!}\cdot 3 ^{3}= \frac{8!}{3!; 5!}\cdot 27= \frac{40,320}{6\cdot 120}\cdot 27 = 1,512$

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Question

Simplify the expression:

$(0.6x + 0.7) ^{3}$

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Answer

You can use the pattern for cubing a binomial sum, setting :

$\left (A + B \right )^{3} = A^{3} + 3A ^{2}B + 3AB^{2} + B ^{3}$

$(0.6x + 0.7) ^{3} =\left ( 0.6x \right )^{3} + 3 \cdot \left ( 0.6x \right ) ^{2} \cdot 0.7 + 3\cdot 0.6x \cdot 0.7 ^{2} + 0.7 ^{3}$

$=0.216x^{3} + 0.756x ^{2} + 0.882x + 0.343$

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Question

Simplify:

$\left ( 24x ^{3}-3x^{2}+ 12x - 18\right ) \div 6x$

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Answer

Rewrite, distribute, and simplify where possible:

$\left ( 24x ^{3}-3x^{2}+ 12x - 18\right ) \div 6x$

$=\frac{ 24x ^{3}-3x^{2}+ 12x - 18 }{6x}$

$=\frac{ 24x ^{3}}{6x} - \frac{ 3x^{2} }{6x} + \frac{ 12x }{6x} - \frac{ 18 }{6x}$

$=\frac{ 24}{6} \cdot x ^{3-1} - \frac{ 3 }{6}\cdot x^{2-1} + \frac{ 12 }{6 } - \frac{ 18 }{6 } \cdot \frac{1}{x}$

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

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

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Question

If positive integer N is divided by 24, the remainder is 6. What is the remainder when N is divided by 10?

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Answer

The simplest way to solve this problem is to start by picking the smallest positive integer that can be divided by 24. That would be 24, since $24\ast1=24$ . Then, since we need a number that when divided by 24, leaves a remainder of 6, simply add 6 to 24. That gives us 30.

30 is the smallest positive integer that leaves a remainder of 6 when divided by 24.

Finally, divide 30 by 10.

The remainder is 0.

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Question

X and Y are positive integers, such that . Which of following numbers could be the remainder of ?

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Answer

Let's set up the problem using algebraic symbols.

,

where Q is the quotient of the answer, and r is the remainder.

$x/y=37.14=37+\frac{14}{100}=37+\frac{7}{50}$ ,

which means that $r/y=\frac{7}{50}$ .

Hence, the remainder MUST be a multiple of 7. The only multiple of 7 in the answer choices is 21, so that is our answer.

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Question

A positive integer, X, leaves a remainder of 3 when divided by 5, and a remainder of 4 when divided by 8. What is the remainder when X is divided by 9?

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Answer

We want to find the smallest number that fits initial conditions, so we need to write two equations to represent the two conditions. If X leaves a remainder of 3 when divided by 5, that is equivalent to:

, where is some integer greater than or equal to 0.

Likewise, the second condition can be expressed by:

, where is some integer greater than or equal to 0.

For the first equation, X could be 3, 8, 13, 18, 23, 28, 33, 38...

For the second equation, X could be 4, 12, 20, 28, 36, 42...

The first number that fits our conditions is 28.

So, finally, what's the remainder when 28 is divided by 9?

The answer is 1.

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