PSAT Math : How to divide polynomials

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : Polynomial Operations

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

 

Possible Answers:

5

6

4

3

Correct answer:

3

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. 

 

 

Example Question #2 : Polynomial Operations

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

II & III

I and II

I and III

I, II and 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

Example Question #1 : How To Divide Polynomials

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:

6

4

3

5

2

Correct answer:

2

Explanation:

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

Then 3a + 5, where = 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,

/ 7 = b  4/7

/ 7 = (7b + 4) / 7

a = (7b + 4)

 

then 3a + 5 = 3 (7+ 4) + 5

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

= (7+ 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.

Example Question #21 : Variables

 

 

Polydivision1

Possible Answers:

36

42

100

45

38

Correct answer:

42

Explanation:

Polydivision2

 

 Polydivision4

Example Question #1 : How To Divide Polynomials

What is the remainder when the polynomial \(\displaystyle x^{4} - 5x^{3} + 3x - 17\) is divided by \(\displaystyle x - 6\) ?

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle 215\)

\(\displaystyle 2,341\)

\(\displaystyle 217\)

\(\displaystyle 2,411\)

Correct answer:

\(\displaystyle 217\)

Explanation:

By the remainder theorem, if a polynomial \(\displaystyle p (x)\) is divided by the linear binomial \(\displaystyle x - c\), the remainder is \(\displaystyle p(c)\) - that is, the polynomial evaluated at \(\displaystyle x = c\). The remainder of dividing \(\displaystyle x^{4} - 5x^{3} + 3x - 17\) by \(\displaystyle x - 6\) is the dividend evaluated at \(\displaystyle x = 6\), which is

\(\displaystyle x^{4} - 5x^{3} + 3x - 17\)

\(\displaystyle = 6^{4} - 5 \cdot 6^{3} + 3 \cdot 6 - 17\)

\(\displaystyle = 1,296 - 5 \cdot 216 + 18 - 17\)

\(\displaystyle = 1,296 - 1,080 + 18 - 17\)

\(\displaystyle = 217\)

Example Question #3 : Multiplying And Dividing Polynomials

Simplify: \(\displaystyle \frac{6x^7y^3z^9}{3x^6y^3z}\)

 

Possible Answers:

\(\displaystyle 6xyz^8\)

\(\displaystyle 2xz^8\)

\(\displaystyle xyz\)

\(\displaystyle xz^8\)

\(\displaystyle xyz^8\)

Correct answer:

\(\displaystyle 2xz^8\)

Explanation:

Cancel by subtracting the exponents of like terms:

\(\displaystyle \frac{6x^7y^3z^9}{3x^6y^3z} = 2x^{7-6}y^{3-3}z^{9-1}=2xy^0z^8=2xz^8\)

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