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PSAT Math : Variables

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Example Questions

Example Question #21 : Variables

Which of the following monomials has degree 999?

Possible Answers:

$444x^{999}yz$

$333x^{999}y^{999}z^{999}$

$999x^{4}y^{5}z^{6}$

None of the other responses is correct.

$100x^{333}y^{333}z^{333}$

Correct answer:

$100x^{333}y^{333}z^{333}$

Explanation:

The degree of a monomial term is the sum of the exponents of its variables, with the default being 1.

For each monomial, this sum - and the degree - is as follows:

$444x^{999}yz$ : 999 + 1 + 1 = 1,001

$333x^{999}y^{999}z^{999}$ : 999+999+999 = 2,997

$999x^{4}y^{5}z^{6}$ : 4+5+6 = 15 (note - 999 is the coefficient)

$100x^{333}y^{333}z^{333}$ : 333+333+333 = 999

$100x^{333}y^{333}z^{333}$ is the correct choice.

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Example Question #22 : Variables

Find the degree of the polynomial

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

Possible Answers:

$\small 3$

None of the other answers

$\small 4$

$\small 5$

Correct answer:

$\small 5$

Explanation:

The degree of the polynomial is the largest degree of any one of it's individual terms.

$\small - x^{2}+3x - 2x^{5} - x^{3} +4$

The degree of $\small -x^2$ is $\small 2$

The degree of $\small 3x$ is $\small 1$

The degree of $\small -2x^5$ is $\small 5$

The degree of $\small -x^3$ is $\small 3$

The degree of $\small 4$ is $\small 0$

$\small 5$ is the largest degree of any one of the terms of the polynomial, and so the degree of the polynomial is $\small 5$ .

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Example Question #1 : Polynomials

Add the polynomials.

$(x^{2}+5x+12) + (3x^{3}+3x+8)$

Possible Answers:

$3x^{3}+x^{2}+6x+17$

$4x^{2}+8x+20$

$3x^{3}+x^{2}+8x+20$

$4x^{3}+2x^{2}+4x+14$

Correct answer:

$3x^{3}+x^{2}+8x+20$

Explanation:

We can add together each of the terms of the polynomial which have the same degree for our variable. $(3x^{3}) + (x^{2}) + (5x+3x) + (12+8) = 3x^{3}+x^{2}+8x+20$

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Example Question #21 : Variables

(x^4+7x^2-5x+4)-(-4x^4+5x^3-x^2+3)

Possible Answers:

5x^4-5x^3+8x^2-5x+1

5x^4+5x^3+6x^2-5x+7

x^4+7x^2-10x+1

x^4-5x^3+11x^2-5x+1

-3x^4+5x^3+6x^2-5x+7

Correct answer:

5x^4-5x^3+8x^2-5x+1

Explanation:

Step 1: Distribute the negative to the second polynomial:

(x^4+7x^2-5x+4)-(-4x^4+5x^3-x^2+3)

x^4+7x^2-5x+4+4x^2-5x^3+x^2-3

Step 2: Combine like terms:

x^4+4x^4-5x^3+7x^2+x^2-5x+4-3

5x^4-5x^3+8x^2-5x+1

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Example Question #2 : Polynomials

$F(x) = x^{3} + x^{2} - x + 2$

and

$G(x) = x^{2} + 5$

What is FG(x) ?

Possible Answers:

$(FG)(x) = x^{3} - x - 3$

$(FG)(x) = x^{5} + x^{4} - x^{3} + 2x^{2} - 5x -10$

$(FG)(x) = x^{5} + x^{4} - x - 2$

$(FG)(x) = x^{3} + 2x^{2} - x + 7$

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

Correct answer:

$(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10$

Explanation:

$(FG)(x) = F(x)G(x)$ so we multiply the two function to get the answer. We use $x^{m}x^{n} = x^{m+n}$

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Example Question #1 : Multiplying And Dividing Polynomials

Multiply:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

Possible Answers:

$1,000Y^{3} + 343$

$1,000Y^{3} + 700Y^{2}- 490 Y- 343$

$1,000Y^{3}+ 100Y^{2}-49 Y- 343$

$1,000Y^{3} - 700Y^{2}+ 490 Y- 343$

$1,000Y^{3} - 100Y^{2}+49 Y- 343$

Correct answer:

$1,000Y^{3} + 343$

Explanation:

$\left (100Y^{2}- 70Y + 49 \right )(10Y+7)$

$= \left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

This product fits the sum of cubes pattern, where A = 10Y, B = 7 :

$(A^{2}-AB +B^{2})(A+B) = A^{3}+B^{3}$

$\left [\left (10Y \right )^{2}- 10Y\cdot 7 +7^{2} \right ](10Y+7)$

$= (10Y)^{3}+7^{3} = 1,000Y^{3} + 343$

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Example Question #1 : Polynomials

If 3 less than 15 is equal to 2x, then 24/x must be greater than

Possible Answers:

Correct answer:

Explanation:

Set up an equation for the sentence: 15 – 3 = 2x and solve for x. X equals 6. If you plug in 6 for x in the expression 24/x, you get 24/6 = 4. 4 is only choice greater than a.

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Example Question #2 : Polynomials

Given a♦b = (a+b)/(a-b) and b♦a = (b+a)/(b-a), which of the following statement(s) is(are) true:

I. a♦b = -(b♦a)

II. (a♦b)(b♦a) = (a♦b)²

III. a♦b + b♦a = 0

Possible Answers:

I only

I and III

I, II and III

I and II

II & III

Correct answer:

I and III

Explanation:

Notice that - (a-b) = b-a, so statement I & III are true after substituting the expression. Substitute the expression for statement II gives ((a+b)/(a-b))((a+b)/(b-a))=((a+b)(b+a))/((-1)(a-b)(a-b))=-1 〖(a+b)〗²/〖(a-b)〗² =-((a+b)/(a-b))² = -(a♦b)² ≠ (a♦b)²

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Example Question #11 : Polynomial Operations

If a positive integer a is divided by 7, the remainder is 4. What is the remainder if 3a + 5 is divided by 3?

Possible Answers:

Correct answer:

Explanation:

The best way to solve this problem is to plug in an appropriate value for a. For example, plug-in 11 for a because 11 divided by 7 will give us a remainder of 4.

Then 3a + 5, where a = 11, gives us 38. Then 38 divided by 3 gives a remainder of 2.

The algebra method is as follows:

a divided by 7 gives us some positive integer b, with a remainder of 4.

Thus,

a / 7 = b 4/7

a / 7 = (7b + 4) / 7

a = (7b + 4)

then 3a + 5 = 3 (7b + 4) + 5

(3a+5)/3 = [3(7b + 4) + 5] / 3

= (7b + 4) + 5/3

The first half of this expression (7b + 4) is a positive integer, but the second half of this expression (5/3) gives us a remainder of 2.

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Example Question #1 : How To Divide Polynomials

Polydivision1

Possible Answers:

100

Correct answer:

Explanation:

Polydivision2

Polydivision4

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PSAT Math : Variables

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #21 : Variables

Example Question #22 : Variables

Example Question #1 : Polynomials

Example Question #21 : Variables

Example Question #2 : Polynomials

Example Question #1 : Multiplying And Dividing Polynomials

Example Question #1 : Polynomials

Example Question #2 : Polynomials

Example Question #11 : Polynomial Operations

Example Question #1 : How To Divide Polynomials

All PSAT Math Resources

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