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MATHEMATICS > Polynomials > Flashcards

Polynomials Flashcards

1
Q

an algebraic expression where the exponents of its variables are whole numbers.

A

Polynomials

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2
Q

What expressions are NOT polynomials?

A
  1. Expressions with fractional or decimal exponents in the variable are not polynomials

For example, 4x1/2+ 2y3is not a polynomial since one of its variables, which is x, has a fractional exponent of ½.

  1. Expressions with negative exponents in the variable are not polynomials

For example, 2a-3b – 5a2b3+ ab is not a polynomial since one of its variables, which is a, has a negative exponent which is – 3.

  1. Expressions with variables in the denominator are not polynomials

For example, 3x – 2⁄y is not a polynomial since it has a variable (which is y) in the denominator.

How about x + y⁄2? Is this a polynomial?

Although 2 is the denominator, 2 is not avariable. This means that x + y⁄2 can be considered a polynomial since it has no variable in the denominator.

Why does a variable in the denominator disqualify an algebraic expression as a polynomial?

As per thenegative exponent rule, if a variable is raised to a negative exponent, we should put that variable in the denominator so that the variable will now have a positive exponent.

If a variable is in the denominator, then it implies that before the negative exponent rule was applied, the variable had a negative exponent in the numerator.

We know that a negative exponent in the variable makes an expression a non-polynomial. This is the reason why variables in the denominator make an expression non-polynomial.

  1. Expressions with variables under the radical sign are not polynomials

Square root (√) and cube root (∛) are some of the examples of radical signs.

As an example, let’s consider the expression √x – y. Since it has a variable (which is x) that is under the radical sign, then √x – y is not a polynomial.

How about √2 + x? Is this a polynomial?

Look at the radical sign. Note that 2 is inside the radical sign. 2 is a constant and not a variable. Thus, we can consider √2 + x as a polynomial

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3
Q

____in a polynomial consists of a number multiplied by a variable with a whole number exponent. The constant part is also a term of the polynomial.

A

Term

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4
Q

Example:Determine the terms in 3x2+ 5y – 2xz

A

Solution:The terms in the given polynomials are 3x2, 5y, and 2xz.

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5
Q

Two or more terms are ____ terms if their variables and exponents (of the variables) are the same

A

Like terms

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6
Q

Example 1:Are 2xz and 5xz like terms?

A

Solution:Yes, because these terms have the same variables (which arexandz).

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7
Q

Example 2:Which of the following does not belong to the group of like terms: 5a2b, -4a2b, 3a2b2, and 9a2b.

A

Solution:3a2b2does not belong to the group because it has a different exponent for its variable b.

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8
Q

the highest exponent of the variable of a polynomial.

A

Degree of a polynomial

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9
Q

Example:What is the degree of the polynomial 3k^7– 5k^2+ k – 9?

A

Solution:The highest exponent of the variable in the polynomial is 7. Thus, the degree of the polynomial is 7.

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10
Q

A polynomial with 1 term

A

Monomial

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11
Q

A polynomial with 2 terms

A

Binomial

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12
Q

A polynomial with 3 terms

A

Trinomial

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13
Q

A polynomial with 4 terms and above

A

Multinomial

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14
Q

an expression that has one term. This means that a constant, a variable, or a product of a constant and a variable with an exponent is a monomial. Examples: 4, a, 5x, -9y2, and 4a2b.

A

Monomial

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15
Q

an expression that has two terms.Examples: x + 2, 2y + z, 4ab – 3b2, and p2q – 3.

A

Binomial

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16
Q

an expression that has three terms. Examples: x2+ 2xy + y2, 3x – y + 5, and 6x3y + y – 9

A

Trinomial

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17
Q

an expression with more than three terms. Examples: a + 2ab + 3abc + 4bcd and 4x2yz + xy3z – xyz2+ xyz + 1.

A

Multinomial

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18
Q

Degree 0

A

Constant

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19
Q

Degree 1

A

Linear

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20
Q

Degree 2

A

Quadratic

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21
Q

Degree 3

A

Cubic

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22
Q

Degree 4

A

Quartic

23
Q

Degree 5

A

Quintic

24
Q

a polynomial with a degree of 0. For example, the classification of 9 according to its degree is constant since the highest exponent it has is 0. Note that we can express 9 as 9x0.

A

Constant

25
Q

a polynomial with a degree of 1. For example, x + y is linear since the highest exponent of its variables is 1. Also, 3x + 9 is linear since its largest exponent of the variable is 1.

A

Linear

26
Q

– a polynomial with a degree of 2. For example, x2+ 2x + 1 is ___ since the highest exponent of its variable is 2

A

Quadratic

27
Q

a polynomial with a degree of 3. For example, x3+ 3x2+ 3x + 1 is _____ since the highest exponent of its variable is 3.

A

Cubic

28
Q

a polynomial with a degree of 4.

A

Quartic

29
Q

a polynomial with a degree of 5

A

Quintic

30
Q

If we express a polynomial in its standard form, the first term of that polynomial is called

A

Leading term

31
Q

Example 1:Write 8y^2+ y^3– 4y + y^5in standard form.

A

Solution:We just write the terms of the polynomial in a manner such that the exponents of the variables are in decreasing order.Thus, the standard form of the given polynomial is y^5+ y^3+ 8y^2– 4y.

32
Q

Example 2:What is the leading term of 5z^8+ 2z + z^7– 2?

A

Solution:To determine the leading term of this polynomial, we should write it first in standard form. If we express the polynomial in standard form, we have 5z8+ z7+ 2z – 2.The first term of this polynomial in standard form is 5z^8.Therefore, 5z^8is the leading term.

33
Q

To learn how to add polynomials, follow the steps below:

A

Arrange the given polynomials in standard formPlace the like terms of the given polynomials in columnsAdd the like terms

34
Q

Example 1:Add 5x^2+ 3x + 2 and 2x^2+ x – 1

A

Solution:

Step 1: Arrange the given polynomials in standard form. Note that the given polynomials are already written in standard form since the exponents of their variables are in decreasing order. Hence, we can skip this step.

Step 2:Place like terms of the given polynomials in columns. Recall that the terms of given polynomials are like terms if the variables and the exponents of these variables are the same. Recall also that the numerical coefficients of like terms can be different.

Step 3:Add the like terms. To add like terms, we add the numerical coefficients and then copy the common variable and the exponent of it.

Thus, the answer is7x^2+ 4x + 1.

35
Q

Example 2:What is the sum if you add y^2– 2z + x^3,5z – 3y + x^2,and2x^3+ 5y^2?

A

Solution:

Step 1: Arrange the given polynomials in standard form. If we arrange the given polynomials into standard form, we will have the following:

x3+ y2– 2z

x2– 3y + 5z

2x3+ 5y2

Step 2:Place the like terms of the given polynomials in columns.

Step 3: Add the like terms.

Therefore, the sum is3x^3+ x^2+ 6y^2– 3y + 3z

36
Q

Example 3:Add 62xy – 5x^2y + 3 by -2x^2y + 10xy – y + 5

A

Solution:

Step 1: Arrange the given polynomials in standard form. If we arrange the given polynomials into standard form, we will have the following:

-5x2y + 62xy + 3

-2x2y + 10xy – y + 5

Step 2:Place the like terms of the given polynomials in columns.

Step 3: Add the like terms. Do not forget the rules onoperations on integerswhen dealing with signed numbers.

The answer is-7x^2y + 72xy – y + 8.

37
Q

To know how to subtract polynomials, follow these steps:

A

Write the given polynomials in standard formChange the sign into addition and reverse the sign of each term of the subtrahend (or the second polynomial)
Add the polynomials

38
Q

Example 1:Subtract 8x^2– 2x + 1 by 2x^2– 5x – 7

A

Solution:

Step 1: Write the given polynomials in standard form. The given polynomials are already in standard form since their terms are already arranged based on the decreasing exponent of the variables.

Step 2: Change the sign into addition and reverse the sign of each term of the subtrahend (or the second polynomial). The subtrahend in the given problem is 2x2– 5x – 7. If we reverse the sign of each term of this polynomial, we have -2x2+ 5x + 7.

Step 3:Add the polynomials. We are now going to add 8x2– 2x + 1 to the polynomial we have obtained from step 2:

Hence, the answer is6x^2+ 3x + 8.

39
Q

Example 2:What is the difference between 7x^5– 5y + z^2and 5x^3+ 2x^5– 6x – 2z^2?

A

Solution:

Step 1: Write the given polynomials into standard form. If we arrange the given polynomials into standard form, we will obtain the following:

7x5+ z2– 5y

2x5+ 5x3– 2z2– 6x

Step 2: Change the sign into addition and reverse the sign of each term of the subtrahend (or the second polynomial). If we reverse the signs of each term of the second polynomial, we will obtain: -2x5– 5x3+ 2z2+ 6x

Step 3: Add the polynomials. We are now going to add 7x5+ z2– 5y to the polynomial we have obtained from step 2:

Hence, the answer is5x^5– 5x^3+ 3z^2+ 6x – 5y

40
Q

When we multiply polynomials, we apply the

A

distributive property of multiplication over addition(or simply distributive property

41
Q

tells us that multiplying the sum of two or more addends by a certain number is equal to the result when we multiply each addend by the same number.

A

Distributive property

42
Q

Example:Compute for 4(5 + 9).

A

Solution:If we usePEMDAS, we will obtain the following:

4(5 + 9)

4 (14)

56

Thus, 4(5 + 9) =56.

Now, let’s try to apply the distributive property:

4(5 + 9) = 4(5) + 4(9) = 20 + 36 = 56

Thus, by applying the distributive property, we obtain the following: 4(5 + 9) =56.

43
Q

Suppose we have the polynomial 5x^2+ 3x – 1 and we want to multiply it by a monomial like 2x.

A

Let us express our problem above as a mathematical sentence:

(5x2+ 3x – 1)(2x)

Since multiplication is commutative (changing the position of numbers in a multiplication process will not change the result), we can express it as:

(2x)(5x2+ 3x – 1)

Take a look at our mathematical sentence above. Notice that we are multiplying a certain quantity (2x)to a sum of addends (5x2+ 3x – 1).This means that we can apply the distributive property.

Applying the distributive property:

After we “distribute” 2x to the addends, we will perform multiplication. Take note that we apply thelaws of exponents (the product rule, in particular)when we multiply the same variables.

Therefore, (2x)(5x2+ 3x – 1) =10x^3+ 6x^2– 2x

44
Q

Example:Multiply 8ab + 2a – 3c by 4ab

A

Solution:We have (4ab)(8ab + 2a – 3c). Applying the distributive property:

(4ab)(8ab) +(4ab)(2a) – (4ab)(3c)

Performing multiplication to each terms:

32a2b2+ 8a2b – 12abc

Therefore, the answer is32a^2b^2+ 8a^2b – 12abc

45
Q

Suppose we have the polynomial 3p2+ 2p – 1 and we want it to be multiplied by a binomial such as 2p + 1.How can we multiply these expressions?

A

If we can express our given problem as a mathematical sentence, we have:

(2p + 1)(3p2+ 2p – 1)

In this case, we can apply the distributive property. We can distribute the first term of the binomial which is 2p to 3p2+ 2p – 1 and we can also distribute the second term of the binomial which is 1 to the same polynomial (i.e., 3p2+ 2p – 1).

After we have distributed the terms of the binomial to the addends of the polynomial, we can apply again the distributive property:

Perform multiplication to each term and apply the product rule.

Note that we can combine some of the like terms of the resulting polynomial:

Therefore, the answer is6p^3+ 7p^2– 1

46
Q

Example:Multiply 5a^2– 3ab + 2 by a – 2b

A

Solution:We have: (a – 2b)(5a2– 3ab + 2). Distributing each terms of the binomial to the polynomial:

a(5a2– 3ab + 2) – 2b(5a2– 3ab + 2)

Applying the distributive property:

[a(5a2) – a(3ab) + a(2)] – [2b(5a2) – 2b(3ab) + 2b(2)]

Multiplying each terms:

[5a3– 3a2b + 2a] – [10a2b – 6ab2+ 4b]

We can rewrite the expression above as:

[5a3– 3a2b + 2a] + [-10a2b + 6ab2– 4b]

Combining like terms:

5a3– 13a2b + 6ab2+ 2a – 4b

Therefore, the answer is5a^3– 13a^2b – 6ab^2+ 2a – 4b

47
Q

Example 1:Multiply 3a^2+ 2a – 1 by a^3– 4a + 1.

A

Solution:Expressing the mathematical problem above as a mathematical sentence, we have:

(3a2+ 2a – 1)(a3– 4a + 1)

We then multiply each term of 3a2+ 2a – 1 to a3– 4a + 1

Applying the principle of distributive property:

Putting together the terms and combining like terms:

Therefore, the answer is3a^5+ 2a^4– 13a^3– 5a^2+ 6a – 1.

48
Q

Example 2:Multiply 5x^2y + y + 2 by y^2+ 3x – 1

A

Solution:Expressing the mathematical problem above as a mathematical sentence, we have:

(5x2y + y + 2)(y2+ 3x – 1)

We multiply each term of 5x2y + y + 2 to y2+ 3x – 1:

5x2y(y2+ 3x – 1) + y(y2+ 3x – 1) + 2(y2+ 3x – 1)

Applying the distributive property:

5x2y(y2+ 3x – 1) + y(y2+ 3x – 1) + 2(y2+ 3x – 1)

(5x2y3+ 15x3y – 5x2y) + (y3+ 3xy – y) + (2y2+ 6x – 2)

Combining like terms and writing the resulting polynomial in standard form:

(5x2y3+ 15x3y – 5x2y) + (y3+ 3xy – y) + (2y2+ 6x – 2)

5x2y3+ 15x3y – 5x2y + y3+ 2y2+ 3xy + 6x – y – 2

Therefore the answer is5x^2y^3+ 15x^3y – 5x^2y + y^3+ 2y^2+ 3xy + 6x – y – 2

49
Q

Suppose we want to divide 25x2+ 10x – 15 by 5x + 5

25x^2+ 10x – 15 ÷ (5x + 5)

A

The first thing we have to do is put 25x2+ 10x – 15 inside the division bracket and then put 5x + 5 outside of it.

Next, we divide the first term of the polynomial inside the division bracket by the first term of the polynomial outside the division bracket. To perform this, we apply thequotient rule. We put the result above the bracket and align it to the first term of the polynomial inside the bracket.

We divide 25x2by 5x and obtain 5x. We then put this answer (i.e., 5x) above the division bracket, making sure it’s aligned to the first term of the dividend.

Then, we multiply the divisor by the answer we obtained earlier and subtract the answer from the dividend.

This means that we multiply 5x + 5 by 5x to obtain 25x2+ 25x.We then subtract 25x2+ 25x from 25x2+ 10x to obtain -15x.

We bring down – 15 to create a new polynomial.

We repeat the things we performed above.

Hence, the answer is 5x – 3.

50
Q

1) Which of the following is quadratic?

a) 3x - 5
b) x^2 + 6x - 1
c) 2x^2y + xy
d) -4a^2- 3a^3

A

B

51
Q

2) Multiply -2x by (x - 9)

a) -2x^2 + 18x
b) -2x^2- 18x
c) 2x^2- 18x
d) None of the above

A

A

52
Q

3) Add 3xy^2 + 2x + z and xy^2- z
a) 4x^2y + 2z
b) 4x^2y^2- 2z
c) 4xy^2 + 2x
d) 4xy^2 + 2x - z

A

C

53
Q

4) Write -3x^4y^2 + 2x^2y^3- x^3y^4 in standard form
a) -x^3y^4- 3x^4y^2 + 2x^2y^3
b) -3x^4y^2 - x^3y^4 + 2x^2y^3
c) 2x^2y^3- 3x^4y^2- x^3y^4
d) None of the above

A

A

54
Q

5) What is the smallest possible whole number value of k so that 2x^3 + 4x^k- 3 = 0 has a degree
that is higher than that of a cubic polynomial?
a) 1
b) 2
c) 3
d) 4

A

D

MATHEMATICS (21 decks)

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