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PSAT Math : Algebra

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Example Question #71 : Algebra

varies inversely as both the square of and the square root of . Assuming that does not depend on any other variable, which statement is true of concerning its relationship to ?

Possible Answers:

varies inversely as the fourth power of .

varies directly as the fourth power of .

varies inversely as .

varies directly as the fourth root of .

varies inversely as the fourth root of .

Correct answer:

varies inversely as the fourth power of .

Explanation:

varies inversely as both the square of and the square root of , meaning that for some constant of variation ,

$xy^{2}\sqrt{z}= K$ .

Square both sides, and the expression becomes

$x^{2}y^{4}z= K^{2}$

$K'= K^{2}$ takes the role of the new constant of variation here, and we now have

$zx^{2}y^{4} = K'$ ,

meaning that varies inversely as the fourth power of .

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Example Question #71 : Algebra

varies directly as the square of and inversely as ; $A= 4 \pi r^{2}$ and $V = \frac{4}{3} \pi r^{3}$ . Assuming that does not depend on any other variables, which of the following gives the variation relationship of to ?

Possible Answers:

varies inversely as .

varies directly as the fourth power of .

varies inversely as the seventh power of .

varies inversely as the fourth power of .

varies directly as .

Correct answer:

varies directly as .

Explanation:

varies directly as the square of and inversely as ; therefore, for some constant of variation ,

$\frac{QV }{A^{2}} = K$

Setting $A= 4 \pi r^{2}$ and $V = \frac{4}{3} \pi r^{3}$ , the formula becomes

$\frac{Q \left ( \frac{4}{3}\pi r^{3} \right ) }{(4\pi r^{2})^{2}} = K$

$\frac{Q \left ( \frac{4}{3}\pi r^{3} \right ) }{(16\pi^{2} r^{4})} = K$

$\frac{Q }{12\pi r } = K$

$\frac{Q }{r } = 12 \pi K$

Setting $K' = 12 \pi K$ as the new constant of variation, we have a new variation equation

$\frac{Q }{r } =K'$ ,

meaning that varies directly as .

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Example Question #1 : How To Multiply A Monomial By A Polynomial

If you have a rectangle with a width of and a length of 7n-3y +2 , what is the area of the rectangle?

Possible Answers:

$21n-9y^{2}+6$

$14ny^{3}$

$56n^{2}-27ny+18n$

21ny-9ny+2ny

$21n^{2}-9ny+6n$

Correct answer:

$21n^{2}-9ny+6n$

Explanation:

To find the area of a rectangle, multiply the length times the width. Therefore, you must multiply times 7n-3y +2 . To do that, you must multiply the monomial times each part of the trinomial, like so:

7n(3n)-3y(3n) +2(3n)

21n^2-9ny+6n

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Example Question #2 : How To Multiply A Monomial By A Polynomial

Find the product:

6x^3y^5(3x^2z^4+x^3y+7y)

Possible Answers:

18x^5y^5z^4 + 6x^6y^6+42x^3y^6

9x^5y^6+6x^3y^6+7

6x^3y^5z^4+6x^3y^5+7y^6

18x^6y^5z^4+6x^9y^5+42x^3y^5

Correct answer:

18x^5y^5z^4 + 6x^6y^6+42x^3y^6

Explanation:

Use the distributive property:

6x^3y^5(3x^2z^4+x^3y+7y)

(6x^3y^5)(3x^2z^4)+(6x^3y^5)(x^3y)+(6x^3y^5)(7y)

Simplify: don't forget to use the rules of multiplying exponents (add them)

18x^5y^5z^4 + 6x^6y^6+42x^3y^6

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Example Question #3 : How To Multiply A Monomial By A Polynomial

Find the product:

4a^2(ab^2+a^2b+ac^3)

Possible Answers:

8a^2b^2+4a^4b

4ab^3+4a^2b^3+4ac

4a^3b^2+4a^4b+4a^3b^3

4a^2b^2+4a^4b+4a^2b^3

4a^2b^3+4ab

Correct answer:

4a^3b^2+4a^4b+4a^3b^3

Explanation:

Find the product:

4a^2(ab^2+a^2b+ac^3)

Use the distributive property:

$(4a^2\cdot ab^2)+(4a^2\cdot a^2b)+(4a^2\cdot ac^3)$

When multiplying variables with exponents, add the exponents:

4a^3b^2+4a^4b+4a^3c^3

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Example Question #1 : How To Multiply A Monomial By A Polynomial

Find the product:

5x^3(x^2+2x+4)

Possible Answers:

5x^3+x^2+2x+4

5x^5+10x^4+20x^3

5x^2+2.5x^2+1.25x^3

5x^6+10x^4+20x^3

5x^6+30x^3

Correct answer:

5x^5+10x^4+20x^3

Explanation:

Find the product:

5x^3(x^2+2x+4)

Use the distributive property:

$(5x^3 \cdot x^2)+(5x^3 \cdot 2x)+(5x^3 \cdot 4)$

5x^5+10x^4+20x^3

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Example Question #1 : Distributive Property

$(\sqrt5+\sqrt3)^2=$

Possible Answers:

$2+\sqrt{15}$

$8+\sqrt{15}$

$\sqrt{15}$

$2+2\sqrt{15}$

$8+2\sqrt{15 }$

Correct answer:

$8+2\sqrt{15 }$

Explanation:

$(\sqrt5+\sqrt3)^2=(\sqrt5)^2+2\sqrt5\sqrt3+(\sqrt3)^2=5+2\sqrt{15}+3=8+2\sqrt{15}$

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Example Question #1 : How To Use Foil In The Distributive Property

If (3x - 4)(3x + 4) = 2 , what is the value of 9x^2 ?

Possible Answers:

Correct answer:

Explanation:

Remember that (a – b )(a + b ) = a ²– b ².

We can therefore rewrite (3x – 4)(3x + 4) = 2 as (3x )²– (4)²= 2.

Simplify to find 9x²– 16 = 2.

Adding 16 to each side gives us 9x²= 18.

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Example Question #1 : Distributive Property

If g(x)=2x^2-2 and h(x)=x+4 , then which of the following is equivalent to g(h(x))-h(g(x)) ?

Possible Answers:

-16x-28

2x^3+8x^2-2x-8

16x

16x+28

Correct answer:

16x+28

Explanation:

We are asked to find the difference between g(h(x)) and h(g(x)), where g(x) = 2x² – 2 and h(x) = x + 4. Let's find expressions for both.

g(h(x)) = g(x + 4) = 2(x + 4)² – 2

g(h(x)) = 2(x + 4)(x + 4) – 2

In order to find (x+4)(x+4) we can use the FOIL method.

(x + 4)(x + 4) = x² + 4x + 4x + 16

g(h(x)) = 2(x² + 4x + 4x + 16) – 2

g(h(x)) = 2(x² + 8x + 16) – 2

Distribute and simplify.

g(h(x)) = 2x² + 16x + 32 – 2

g(h(x)) = 2x² + 16x + 30

Now, we need to find h(g(x)).

h(g(x)) = h(2x² – 2) = 2x² – 2 + 4

h(g(x)) = 2x² + 2

Finally, we can find g(h(x)) – h(g(x)).

g(h(x)) – h(g(x)) = 2x² + 16x + 30 – (2x² + 2)

= 2x² + 16x + 30 – 2x² – 2

= 16x + 28

The answer is 16x + 28.

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Example Question #4 : Distributive Property

The sum of two numbers is . The product of the same two numbers is . If the two numbers are each increased by one, the new product is . Find q-p in terms of .

Possible Answers:

s+1

s+2

2s+1

s^2

s-1

Correct answer:

s+1

Explanation:

Let the two numbers be x and y.

x + y = s

xy = p

(x + 1)(y + 1) = q

Expand the last equation:

xy + x + y + 1 = q

Note that both of the first two equations can be substituted into this new equation:

p + s + 1 = q

Solve this equation for q – p by subtracting p from both sides:

s + 1 = q – p

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #71 : Algebra

Example Question #71 : Algebra

Example Question #1 : How To Multiply A Monomial By A Polynomial

Example Question #2 : How To Multiply A Monomial By A Polynomial

Example Question #3 : How To Multiply A Monomial By A Polynomial

Example Question #1 : How To Multiply A Monomial By A Polynomial

Example Question #1 : Distributive Property

Example Question #1 : How To Use Foil In The Distributive Property

Example Question #1 : Distributive Property

Example Question #4 : Distributive Property

All PSAT Math Resources

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