GMAT Math : Understanding factoring

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Example Question #11 : Understanding Factoring

Factor:

$4x^{2}-12xy +9y^{2} - 64x^{2}y^{2}$

Possible Answers:

$\left ( 2x-3y -8xy \right )^{2}$

$\left ( 2x+3y +8xy \right ) \left ( 2x-3y -8xy \right )$

$\left ( 2x-3y +8xy \right ) \left ( 2x-3y -8xy \right )$

$\left ( 2x+3y +8xy \right ) \left ( 2x+3y -8xy \right )$

Correct answer:

$\left ( 2x-3y +8xy \right ) \left ( 2x-3y -8xy \right )$

Explanation:

$4x^{2}-12xy +9y^{2} - 64x^{2}y^{2}$ can be grouped as follows:

$\left (4x^{2}-12xy +9y^{2} \right ) - 64x^{2}y^{2}$

The first three terms form a perfect square trinomial, since $2\cdot \sqrt{4x^{2} } \cdot \sqrt { 9y^{2}} = 2 \cdot 2x \cdot 3y = 12xy$

$4x^{2}-12xy +9y^{2} = \left ( 2x - 3y\right )^{2}$ , so

$\left (4x^{2}-12xy +9y^{2} \right ) - 64x^{2}y^{2}$

$= \left ( 2x - 3y\right )^{2} - 8 ^{2}x^{2}y^{2}$

$= \left ( 2x - 3y\right )^{2} - \left ( 8 xy \right ) ^{2}$

Now use the dfference of squares pattern:

$= \left [ \left ( 2x - 3y \right ) + 8 xy \right ] \left [ \left ( 2x - 3y \right ) - 8 xy \right ]$

$= \left ( 2x - 3y + 8 xy \right ) \left ( 2x - 3y - 8 xy \right )$

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Example Question #1501 : Problem Solving Questions

Solve for when f(x)=0 .

f(x)=x^2+14x+49

Possible Answers:

x=14

x=7

x=49

x=-7

Correct answer:

x=-7

Explanation:

f(x)=x^2+14x+49

0=x^2+14x+49

0=(x+7)(x+7)

x+7=0

x=-7

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Example Question #1502 : Problem Solving Questions

Solve for when f(x)=0 .

f(x)=x^2+x-2

Possible Answers:

and x=-1

x=2 and

x=2 and x=-1

and

Correct answer:

and

Explanation:

f(x)=x^2+x-2

0=x^2+x-2

0=(x+2)(x-1)

x+2=0 and x-1=0

and

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Example Question #1503 : Problem Solving Questions

Solve for :

x^2-8x-9=0

Possible Answers:

x=9 and x=-1

x=9 and x=1

x=-9 and x=-1

x=-9 and x=1

Correct answer:

x=9 and x=-1

Explanation:

x^2-8x-9=0

(x-9)(x+1)=0

x-9=0 and x+1=0

x=9 and x=-1

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Example Question #11 : Understanding Factoring

Factor the expression completely:

$625x^{4} - 450x^{2} + 81$

Possible Answers:

$(5x-3)^{4}$

$(25x^{2}-15x+9)(5x+3)^{2}$

$(25x^{2}-15x+9)(5x-3)^{2}$

$(25x^{2}+9)(5x-3)^{2}$

$(5x+3)^{2}(5x-3)^{2}$

Correct answer:

$(5x+3)^{2}(5x-3)^{2}$

Explanation:

The expression is a perfect square trinomial, as the three terms have the following relationship:

$625x^{4} =\left ( 25x ^{2} \right ) ^{2}$

$81 = 9 ^{2}$

$450x^{2} = 2 \cdot 25x^{2} \cdot 9$

$625x^{4} - 450x^{2} + 81= \left ( 25x ^{2} \right ) ^{2} - 2 \cdot 25x^{2} \cdot 9 + 9 ^{2}$

We can factor this expression by substituting $A = 25x ^{2}, B = 9$ into the following pattern:

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

$\left ( 25x ^{2} \right ) ^{2} - 2 \cdot 25x^{2} \cdot 9 + 9 ^{2} =(25x ^{2}-9)^{2}$

We can factor further by noting that $25x ^{2}-9 = (5x)^{2} - 3^{2}$ , the difference of squares, and subsequently, factoring this as the product of a sum and a difference.

$625x^{4} - 450x^{2} + 81=(25x ^{2}-9)^{2} = \left [ (5x+3)(5x-3) \right ]^{2}$

$625x^{4} - 450x^{2} + 81= (5x+3)^{2}(5x-3)^{2}$

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Example Question #1505 : Problem Solving Questions

If $5(a^{3}+3a^{2}b+3ab^{2}+b^{3})=40$ what is a+b ?

Possible Answers:

$\frac{40}{3}$

$40\tfrac{1}{3}$

Correct answer:

Explanation:

Note that $a^{3}+3a^{2}b+3ab^{2}+b^{3} = (a+b)^{3}$

Therefore $5(a^{3}+3a^{2}b+3ab^{2}+b^{3})=40$ is equivalent to:

$5(a+b)^{3}=40$ $\Leftrightarrow (a+b)^{3}=40/5 = 8 \Leftrightarrow a+b = 2$

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Example Question #11 : Understanding Factoring

Solve for ;

$\frac{x^3-x}{x^3+6x^2+5x}=5$

Possible Answers:

-12/5

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is -13/2 . Our work proceeds as follows:

$\frac{x^3-x}{x^3+6x^2+5x}=5$

$\frac{(x^2-1)(x)}{(x^2+6x+5)(x)} =5$ (Factor an out of the numerator and denominator)

$\frac{(x+1)(x-1)(x)}{(x+1)(x+5)(x)} = 5$ (Factor the quadratic polynomials)

$\frac{x-1}{x+5} =5$ (Cancel common terms)

x-1 = 5(x+5) (Multiply by x+5 to both sides)

x-1 = 5x +25 (Distribute the )

-26 = 4x (Simplify and solve)

$\frac{-13}{2} = x$

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Example Question #1507 : Problem Solving Questions

Solve for by factoring and using the zero product property.

3x^2-74x-25=0

Possible Answers:

$x=-5, x=\frac{5}{3}$

$x=\frac{1}{3},x=-25$

$x=-\frac{1}{3}, x=25$

$x=-1, x=\frac{25}{3}$

Correct answer:

$x=-\frac{1}{3}, x=25$

Explanation:

In order to solve for we must first factor:

3x^2-74x-25=0

(3x+1)(x-25)=0

The zero product property states that if xy=0 then x=0 or y=0 (or both).

Our two equations are then:

(3x+1)=0, (x-25)=0

Solving for in each leaves us with:

3x+1=0

3x=-1

$x=-\frac{1}{3}$

and

x-25=0

x=25

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Example Question #11 : Solving By Factoring

Find the roots of the following function:

f(x)=x^2-3x-18

Possible Answers:

x=-2,x=9

x=-1,x=18

x=-9,x=2

x=-3,x=6

x=-6,x=3

Correct answer:

x=-3,x=6

Explanation:

The roots of a function are the points at which it crosses the x axis, so at these points the value of y, or f(x), is 0. This gives us:

x^2-3x-18=0

So we will have to factor the polynomial in order to solve for the x values at which the function is equal to 0. We need two factors whose product is -18 and whose sum is -3. If we think about our options, 2 and 9 have a product of -18 if one is negative, but there's no way of making these two numbers add up to -3. Next we consider 3 and 6. These numbers have a product of -18 if one is negative, and their sum can also be -3 if the 3 is positive and the 6 is negative. This allows us to write out the following factorization:

x^2-3x-18=0

(x+3)(x-6)=0

x=-3,x=6

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Example Question #11 : Solving By Factoring

Where does the following function cross the -axis?

f(x)=x^2-6x+8

Possible Answers:

x=-2,-4

x=8,-8

x=2,-4

x=-2,4

x=2,4

Correct answer:

x=2,4

Explanation:

We could solve this question a variety of ways. The simplest would be graphing with a calculator, but we will use factoring.

To begin, set our function equal to . We want to find where this function crosses the -axis—in other words, where y=0 .

0=x^2-6x+8

Next, we need to factor the function into two binomial terms. Remember FOIL/box method? We are essentially doing the reverse here. We are looking for something in the form of (x+a)(x+b)=0 .

Recalling a few details will make this easier.

1) a*b must equal positive

2) and must both be negative, because we get positive when we multiply them and when we add them.

3) and must be factors of that add up to . List factors of : 1, 2, 4, 8 . The only pair of those that will add up to are and , so our factored form looks like this:

0=(x-2)(x-4)

Then, due to the zero product property, we know that if x=2 or x=4 one side of the equation will equal , and therefore our answers are positive and positive .

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GMAT Math : Understanding factoring

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Understanding Factoring

Example Question #1501 : Problem Solving Questions

Example Question #1502 : Problem Solving Questions

Example Question #1503 : Problem Solving Questions

Example Question #11 : Understanding Factoring

Example Question #1505 : Problem Solving Questions

Example Question #11 : Understanding Factoring

Example Question #1507 : Problem Solving Questions

Example Question #11 : Solving By Factoring

Example Question #11 : Solving By Factoring

All GMAT Math Resources

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