POLYNOMIALS IN STANDARD FORM WORKSHEET

Problems 1 to 8 : Write each polynomial in standard form. Then, give the leading coefficient.   

Problem 1 :

20y + 4y³ - 2 + y²

Problem 2 :

x³ + x⁵ + 4x

Problem 3 :

5x - 17x⁶

Problem 4 :

-6x⁴ - 5x - 9x³

Problem 5 :

9x³ - x + 3 - 7x⁴

Problem 6 :

x - 9 + √7x³ + 6x²

Problem 7 :

√2x² - (7/2)x⁴ + x - 5x³

Problem 8 :

y² - √5y³ - 11 - (7/3)y + 9y⁴

Problems 9 and 10 : Add the following two polynomials and write the resulting polynomials in standard form. 

Problem 9 :

p(x)  =  6x² - 7x + 2  and  q(x)  =  6x³ - 7x + 15

Problem 10 :

f(x)  =  16x⁴ - 5x² + 9  and  g(x)  =  -6x³ + 7x - 15

Problems 11 and 12 : Find the difference of the following two polynomials and write the resulting polynomials in standard form. 

Problem 11 :

h(x)  =  7x³ - 6x + 1  and  f(x)  =  7x² + 17x - 9

Problem 12 :

h(x)  =  x - 6x³ - 5  and  f(x)  =  12x -7x³ - 5

Detailed Answer Key

Problems 1 to 8 : Write each polynomial in standard form. Then, give the leading coefficient.   

Problem 1 :

20y + 4y³ - 2 + y²

Solution : 

Find the degree of each term :

20y  :  degree 1

4y³  :  degree 3

-2  :  degree 0

y²  :  degree 2

Arrange the terms in order from largest degree to smallest degree.

4y³ + y² + 20y - 2

The standard form is (4y³ + y² + 20y - 2) and the leading coefficient is 4.

Problem 2 :

x³ + x⁵ + 4x

Solution : 

Find the degree of each term :

x³  :  degree 3

x⁵  :  degree 5

4x  :  degree 1

Arrange the terms in order from largest degree to smallest degree.

x⁵ + x³ + 4x

The standard form is (x⁵ + x³ + 4x) and the leading coefficient is 1.

Problem 3 :

5x - 17x⁶

Solution : 

Find the degree of each term :

5x  :  degree 1

-17x⁶  :  degree 6

Arrange the terms in order from largest degree to smallest degree.

-17x⁶ + 5x

The standard form is (-17x⁶ + 5x) and the leading coefficient is -17.

Problem 4 :

-6x⁴ - 5x - 9x³

Solution : 

Find the degree of each term :

-6x⁴  :  degree 4

-5x  :  degree 1

-9x³  :  degree 3

Arrange the terms in order from largest degree to smallest degree.

-6x⁴ - 9x³ - 5x

The standard form is (-6x⁴ - 9x³ - 5x) and the leading coefficient is -6.

Problem 5 :

9x³ - x + 3 - 7x⁴

Solution : 

Find the degree of each term :

9x³  :  degree 3

-x  :  degree 1

3  :  degree 0

-7x⁴  :  degree 4

Arrange the terms in order from largest degree to smallest degree.

-7x⁴ + 9x³ - x + 3

The standard form is (-7x⁴ + 9x³ - x + 3) and the leading coefficient is -7.

Problem 6 :

x - 9 + √7x³ + 6x²

Solution : 

Find the degree of each term :

x  :  degree 1

-9  :  degree 0

√7x³  :  degree 3

6x²  :  degree 2

Arrange the terms in order from largest degree to smallest degree.

√7x³ + 6x² + x - 9

The standard form is (√7x³ + 6x² + x - 9) and the leading coefficient is √7.

Problem 7 :

√2x² - (7/2)x⁴ + x - 5x³

Solution : 

Find the degree of each term :

√2x²  :  degree 2

-(7/2)x⁴  :  degree 4

x  :  degree 1

-5x³  :  degree 3

Arrange the terms in order from largest degree to smallest degree.

-(7/2)x⁴ - 5x³ + √2x² + x

The standard form is (-(7/2)x⁴ - 5x³ + √2x² + x) and the leading coefficient is -7/2.

Problem 8 :

y² - √5y³ - 11 - (7/3)y + 9y⁴

Solution : 

Find the degree of each term :

y²  :  degree 2

-√5y³  :  degree 3

-11  :  degree 0

-(7/3)y  :  degree 1

9y⁴  :  degree 4

Arrange the terms in order from largest degree to smallest degree.

9y⁴- √5y³ + y² - (7/3) y - 11

The standard form is (9y⁴- √5y³ + y² - (7/3) y - 11) and the leading coefficient is 9.

Problems 9 and 10 : Add the following two polynomials and write the resulting polynomials in standard form. 

Problem 9 :

p(x)  =  6x² - 7x + 2  and  q(x)  =  6x³ - 7x + 15

Solution :

p(x) + q(x)  =  (6x² - 7x + 2) + (6x³ - 7x + 15)

=  6x² - 7x + 2 + 6x³ - 7x + 15

Arrange the terms in order from largest degree to smallest degree.

=  6x³ + 6x² - 7x - 7x + 2 + 15

Combine the like terms. 

=  6x³ + 6x² - 14x + 17

Problem 10 :

f(x)  =  16x⁴ - 5x² + 9  and  g(x)  =  -6x³ + 7x - 15

Solution :

f(x) + g(x)  =  (16x⁴ - 5x² + 9) + (-6x³ + 7x - 15)

=  16x⁴ - 5x² + 9 + -6x³ + 7x - 15

Arrange the terms in order from largest degree to smallest degree.

=  16x⁴ - 6x³  - 5x² + 7x + 9 - 15

Combine the like terms. 

=  16x⁴- 6x³  - 5x² + 7x  - 6

Problems 11 and 12 : Find the difference of the following two polynomials and write the resulting polynomials in standard form. 

Problem 11 :

h(x)  =  7x³ - 6x + 1  and  f(x)  =  7x² + 17x - 9

Solution :

h(x) - f(x)  =  (7x³ - 6x + 1) - (7x² + 17x - 9)

=  7x³ - 6x + 1 - 7x² - 17x + 9

Arrange the terms in order from largest degree to smallest degree.

=  7x³ - 7x²- 6x - 17x + 1 + 9

Combine the like terms. 

=  7x³ - 7x² - 23x + 10

Problem 12 :

h(x)  =  x - 6x³ - 5  and  f(x)  =  12x -7x³ - 5

Solution :

h(x) - f(x)  =  (x - 6x³ - 5) - (12x - 7x³ - 5)

=  x - 6x³ - 5 - 12x + 7x³ + 5

Arrange the terms in order from largest degree to smallest degree.

=  -6x³ + 7x³ + x - 12x -5 + 5

Combine the like terms. 

=  x³ - 11x

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