To add or subtract polynomials, simply add or subtract their like terms. To do this, you need add or subtract the coefficients of like terms. When adding two or more polynomials, you need write the terms of the polynomials in a vertical or in a horizontal line, and then combine the like terms. When subtracting the first polynomial from the second polynomial, you must change the sign of the first polynomial. (note: when adding or subtracting polynomials, you will NOT change their exponents)

Practice 15. Add the polynomials.

(a)   (3x⁴ – 12x³ – 7x² + 11x – 21) + (–x⁴ – 2x³ + 9x² – 14x – 33)

(b)   (2x³y² – 2x²y³ – 3xy⁴ + x – 2y) + (–5x³y² + 6x²y³ + xy⁴ – 3x – y)

Solution.

(a)   (3x⁴ – 12x³ – 7x² + 11x – 21) + (–x⁴ – 2x³ + 9x² – 14x – 33)

= 3x⁴ – 12x³ – 7x² + 11x – 21 – x⁴ – 2x³ + 9x² – 14x – 33
To make it easier to see what you are doing, rearrange the polynomials, putting like terms next to each number, being sure to keep the  sign    that is in front of each term with that term.
3x⁴– x⁴– 12x³– 2x³– 7x²+ 9x²+ 11x– 14x– 21– 33
Remember, any time that there is no coefficient in front of a term, it is assumed that the coefficient is 1.
 

= 2x⁴ – 14x³ + 2x² – 3x – 54

(b)   (2x³y² – 2x²y³ – 3xy⁴ + x – 2y) + (–5x³y² + 6x²y³ + xy⁴ – 3x – y)

= 2x³y² – 2x²y³ – 3xy⁴ + x – 2y – 5x³y² + 6x²y³ + xy⁴ – 3x – y

= – 3x³y² + 4x²y³ – 2xy⁴ – 2x – 3y

 (c) Sometimes one of the terms has no match. In that case, assume that you are adding zero to the term that has no match.

Practice 16. Subtract the polynomials. To subtract polynomials, you must change every sign inside the parentheses. You can think of it as multiplying every term inside the parentheses by -1. After you have done that, you can treat the result as an addition problem.

(a)   (4m⁵ – 3m³ + 11m² – mn) – (–m⁵ + m³ – 4m² + 2mn)

(b)   (4a³b – 2a³b² +  11a³b³ – 8ab) – (–a³b + a³b² –  3a³b³ – 4ab)

Solution.

(a)   (4m⁵ – 3m³ + 11m² – mn) – (–m⁵ + m³ – 4m² + 2mn)

   = 4m⁵ – 3m³ + 11m² – mn + m⁵ – m³ + 4m² – 2mn

  = 4m⁵+ m⁵– 3m³– m³+ 11m²+ 4m²– mn– 2mn

   = 5m⁵ – 4m³ + 15m² – 3mn

(b)   (4a³b – 2a³b² +  11a³b³ – 8ab) – (–a³b + a³b² –  3a³b³ – 4ab)

= 4a³b – 2a³b² + 11a³b³ – 8ab + a³b – a³b² + 3a³b³ + 4ab

= 4a³b+ a³b– 2a³b²– a³b²+ 11a³b³+ 3a³b³– 8ab+ 4ab

= 5a³b – 3a³b² + 14a³b³ – 4ab

Practice 17. Perform the indicated operations.

(a)   (4x⁴ – 3x³ – 7x) + (–3x⁴ + 4x³ – 7x²) – (8x³ + 2x² – 5x)

(b)   (–4m⁵ – n³ – m – n) + (–3m⁵ + 4n³ – 3m + 7n) – (8x³ + 2x² – 5x)

Solution.

(a)   (4x⁴ – 3x³ – 7x) + (–3x⁴ + 4x³ – 7x²) – (8x³ + 2x² – 5x)

= 4x⁴ – 3x³ – 7x – 3x⁴ + 4x³ – 7x² – 8x³ – 2x² + 5x

= 4x⁴– 3x⁴– 3x³+ 4x³– 8x³– 7x²– 2x²– 7x+ 5x

(note: Here we kept the exponents in descending order, going from largest to smallest. This is a convention that enables us to know that we have found all the like terms and everyone’s answers will be identical and easier to understand and correct if necessary)
 

= x⁴ – 7x³ – 9x² – 2x

(b)   (–4m⁵ – n³ – m – n) + (–3m⁵ + 4n³ – 3m + 7n) – (8x³ + 2x² – 5x)

      = –4m⁵ – n³ – m – n – 3m⁵ + 4n³ – 3m + 7n – 8x³ – 2x² + 5x

      
       = –4m⁵– 3m⁵– n³+ 4n³– m– 3m– n+ 7n– 8x³ – 2x² + 5x

      = –7m⁵ + 3n³ – 4m + 6n – 8x³ – 2x² + 5x

Real Life Application. The total revenue of computer company is modeled by the expression below.

(–0.06x² + 114.8x)

In this expression, x is the number of computers produced per week and the expression is in dollars.

The total cost of producing x computers is modeled by the expression below.

(0.00007x³ – 0.03x² + 1200x + 14600)

Find a model that represents the profit of the producer.

Solution. To find the profit, simply subtract the cost model from the model of revenue.

(–0.06x² + 114.8x) – (0.00007x³ – 0.03x² + 1200x + 14600) =

                          = –0.06x² + 114.8x – 0.00007x³ + 0.03x² – 1200x – 14600

                           = – 0.00007x³–0.06x²+ 0.03x²+ 114.8x– 1200x– 14600

                          = – 0.00007x³ – 0.03x² – 1085.2x – 14600

Real Life Application 2. In Figure 6.5, the colored region is the side walk around a pool. The dimensions are in feet. Find the area of this region.

Figure 6.5

Solution. To find the area of the side walk, we must subtract the area of the inner rectangle from the area of the outer rectangle.

Area of Outer Rectangle = x (x + y)

                                                = x(x) + x(y)

                                                = x² + xy

Area of Inner Rectangle = y(2x – y)

                                           = y(2x) – y(y)

                                           = 2xy – y²

Area of Side walk = (x² + xy) – (2xy – y²)

                             = x² + xy – 2xy + y²

                             = (x² – xy + y² ) ft²

Practical Exercise 5. Add or subtract.

(a)   (x⁷ + x⁶ – x⁵ + 2x⁴ – x³) – (5x⁶ + x⁵ + x⁴ – 5x³ + 10x²)

(b)   (–m⁵ – n³ – 3m – 2n) + (–m⁵ + n³ – m + n)

Answer

Links for Students, Parents and Teachers

Now let's do Practice Exercise 6-5 (top).

Summary

What is a Monomial? A monomial is an expression which has only one term. Other types of expressions are usually made up of monomials.  

How to Multiply Monomials? Any monomial is built up of three parts: sign, coefficient, and variables. To multiply two monomials, first multiply their signs. To do this, use the rules of product of signs. Then multiply the coefficients. Finally multiply the variables.  

How to Multiply a Monomial by a Binomial? To multiply a monomial by a binomial, use the Distributive Property. Also, you can use the special product identities below to multiply two binomials. 

Special Identities

(a + b)(a + b) = (a + b)² = a² + 2ab + b²

(a – b)(a – b) = (a – b)² = a² – 2ab + b²

(a + b)(a – b) = a² – b²

(a + b)³ = a³ + 3a²b + 3ab² + b³
(a – b)³ = a³ – 3a²b + 3ab² – b³

Dividing Monomials. To divide a monomial by another monomial, follow the steps below.

(1) Divide the signs of the monomials.

(2) Divide the coefficients of the monomials.

(3) Divide the like variables. Use the rule for the division of variables with exponents. To do this, subtract the exponents of like variables.

Scientific Notation. In many subjects, we deal with very small or very large numbers. To present such numbers in an easy way, we use a certain method called scientific notation. Using this way, we can write the distances between galaxies or the mass of an atom in forms that can be read easily.

When is a Number in Scientific Notation? We can write positive numbers in scientific notation. A number in the form of a × 10ⁿ  is in scientific notation. In this form 1< a < 10 and n is an integer. Any number in scientific notation has two parts; a number between 1 and 10, and a power of 10.

Polynomials. Polynomials are algebraic expressions made up of several terms. In any polynomial, the exponents of variables must be whole numbers.

How to Multiply Two Polynomials? To multiply two polynomials, use the Distributive Property several times. Each time take a term from one of the polynomial and multiply it by all the terms of the other polynomial.  

Adding and Subtracting Polynomials. To add or subtract polynomials, simply add or subtract their like terms. To do this, you need add or subtract the coefficients of like terms. When adding two or more polynomials, you need write the terms of the polynomials in vertical or horizontal lines. Then combine the like terms. When subtracting first polynomial from the second polynomial, you must change the sign of the first polynomial.