Algebra 1 : Polynomial Operations

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #22 : How To Find The Degree Of A Polynomial

What is the degree of the following polynomial? 

\displaystyle x^5 + 2y^2 + 38z + 40

Possible Answers:

\displaystyle 2

\displaystyle 4

\displaystyle 40

\displaystyle 5

\displaystyle 38

Correct answer:

\displaystyle 5

Explanation:

To find the degree of a polynomial, simply find the highest exponent in the expression.  As five is the highest exponent above, it is also the degree of the polynomial.

Example Question #23 : How To Find The Degree Of A Polynomial

What is the degree of the following polynomial?  

\displaystyle 2z^3 + 4z^2 + 8z + 5 + 13y^8

Possible Answers:

\displaystyle 3

\displaystyle 8

\displaystyle 8

\displaystyle 2

\displaystyle 5

Correct answer:

\displaystyle 8

Explanation:

To find the degree of a polynomial, simply find the highest exponent in the expression.  As eight is the highest exponent above, it is also the degree of the polynomial.

Example Question #24 : How To Find The Degree Of A Polynomial

What is the degree of the following polynomial?  

\displaystyle 12x^5 + 3x^4 + 2y + 13z -50n^9

Possible Answers:

\displaystyle 12

\displaystyle 5

\displaystyle 4

\displaystyle 13

\displaystyle 9

Correct answer:

\displaystyle 9

Explanation:

To find the degree of a polynomial, simply find the highest exponent in the expression.  As nine is the highest exponent above, it is also the degree of the polynomial.

Example Question #25 : How To Find The Degree Of A Polynomial

What is the degree of the following polynomial?  

\displaystyle 3a + 2b +4c + 5

Possible Answers:

\displaystyle 1

\displaystyle 3

\displaystyle 2

\displaystyle 4

\displaystyle 5

Correct answer:

\displaystyle 1

Explanation:

To find the degree of a polynomial, simply find the highest exponent in the expression.  As one is the highest exponent above (remember that no exponent means that the variable is being raised to one), it is also the degree of the polynomial.

Example Question #26 : How To Find The Degree Of A Polynomial

Find the degree in the following polynomial:

\displaystyle -4x^3 + 2x^2 - 9x + 14

Possible Answers:

\displaystyle 3

\displaystyle 14

\displaystyle -4

\displaystyle 2

\displaystyle 9

Correct answer:

\displaystyle 3

Explanation:

To find the degree of a polynomial, you simply locate the variable with the largest exponent, and that exponent is the degree.  So, in the polynomial

\displaystyle -4x^3 + 2x^2 - 9x + 14

we see that the largest exponent is \displaystyle 3.  So the degree of this polynomial is \displaystyle 3.

Example Question #26 : How To Find The Degree Of A Polynomial

Which of these equations is a third-degree polynomial?

Possible Answers:

\displaystyle y = x^3 + 2x^4 + 2x^5 + 1

\displaystyle y = 5x^2 + 3x^3 + x - 4

\displaystyle y = 3x^4 + 5x^3 + x^2 - x + 5

\displaystyle y = x^2 + x - 5

Correct answer:

\displaystyle y = 5x^2 + 3x^3 + x - 4

Explanation:

The largest exponent on the correct answer and only the correct answer is 3.

Example Question #23 : How To Find The Degree Of A Polynomial

What is the degree of the resulting polynomial once the following expression has been expanded:

\displaystyle (x^{6}+3x)(x^{5}-4x^{2})

Possible Answers:

\displaystyle 30

\displaystyle 8

\displaystyle 6

\displaystyle 19

\displaystyle 11

Correct answer:

\displaystyle 11

Explanation:

The degree of a polynomial is the highest degree that can be found in the expression when it has been expanded out into mononomial components. So first, we need to FOIL. Remember that multiplying exponents means you add them to find the resultant exponent of the new mononomial term: 

\displaystyle (x^{6}+3x)(x^{5}-4x^{2})

First: \displaystyle x^{6} \times x^{5} = x^{11}  

Outer: \displaystyle x^{6} \times -4x^{2}= -4x^{8}

Inner: \displaystyle 3x \times x^{5}= 3x^{6}

Last: \displaystyle 3x \times -4x^{2} = -12x^{3}

Rewrite the entire expresson:

\displaystyle x^{11} - 4x^{8} + 3x^{6} - 12x^{3}

The highest degree is \displaystyle 11, and therefore \displaystyle 11 is the degree of the polynomial.

Example Question #27 : How To Find The Degree Of A Polynomial

Find the degree of the polynomial:

\displaystyle x^3+2x^2+4x+7

Possible Answers:

\displaystyle 7

\displaystyle 2

\displaystyle 3

\displaystyle 4

\displaystyle 1

Correct answer:

\displaystyle 3

Explanation:

Finding the degree of a polynomial is simple - we merely need to see what the largest exponent in the expression is. This will determine the degree of the polynomial. We must keep in mind, however, that coefficients do not matter because they do not influence degree. 

In this problem, the expression is \displaystyle x^3+2x^2+4x+7.

Because the expression is written in descending order for the degree of x, we can quickly see that the largest exponent is 3. Therefore, the degree of this polynomial is 3.

Example Question #30 : How To Find The Degree Of A Polynomial

Find the degree of the polynomial:

\displaystyle \frac{x^8+10x^2+8x^3+27}{x^2}

Possible Answers:

\displaystyle 3

\displaystyle 8

\displaystyle 2

\displaystyle 10

\displaystyle 6

Correct answer:

\displaystyle 6

Explanation:

Finding the degree of a polynomial is simple - we merely need to see what the largest exponent in the expression is. This will determine the degree of the polynomial. We must keep in mind, however, that coefficients do not matter because they do not influence degree. 

In this problem, the expression is \displaystyle \frac{x^8+10x^2+8x^3+27}{x^2}.

In order to see the leading exponent, we must first simplify the expression. Dividing by \displaystyle x^2 leaves us with \displaystyle x^6+10+8x+27x^{-2}

Rearranging the expression so it is written in descending order for the degree of x:

\displaystyle x^6+8x+{27}{x^{-2}}+10

Now we can quickly see that the largest exponent is 6. Therefore, the degree of this polynomial is 6.

 

Example Question #31 : How To Find The Degree Of A Polynomial

Find the degree of the polynomial:

\displaystyle 10x^2+2x^5+7x+x^7+5^{14}

Possible Answers:

\displaystyle 2

\displaystyle 10

\displaystyle 14

\displaystyle 5

\displaystyle 7

Correct answer:

\displaystyle 7

Explanation:

Finding the degree of a polynomial is simple - we merely need to see what the largest exponent in the expression is. This will determine the degree of the polynomial. We must keep in mind, however, that coefficients do not matter because they do not influence degree. 

In this problem, the expression is \displaystyle 10x^2+2x^5+7x+x^7+5^{14}.

Rearranging the expression so it is written in descending order for the degree of x:

\displaystyle x^7+2x^5+10x^2+7x+5^{14}

Now we can easily see that the largest exponent is 7. Therefore, the degree of this polynomial is 7.

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