In algebra, we deal with many different types of expressions. Grouping them helps us to learn rules and concepts easily. One group of expressions is called polynomials. In a polynomial, we have combinations of letters, standing for unknown values that may or may not change according to the rules of mathematic, and numbers. Many of the letters will be raised to powers, such as x², where the 2 is considered a power. When you have an expression such as x², the 2 represents the number of times a letter is multiplied by itself (x times x). These powers are whole numbers. Some special polynomials are named Monomials, Binomials, and Trinomials.

What is a Monomial?

Monomial is a polynomial which has only one term. Here are some examples.

3x

12axy²

–34mnxy²z

1.24abx²y³

 Terms are separated from one another by either a + or – sign.

Practice 1. Which terms are monomials?

(a)   –1212mn + 32

(b)   24

(c)   35abc

(d)

Answer. (b), (c), and (d)

Practice 2. Which of the geometrical formulas below are in the form of monomial?

(a)   Area of Rectangle A = lw

(b)   Area of Square A = s²

(c)   Area of Triangle A = ½ bh

(d)   Area of Rhombus A = lh

(e)   Area of Parallelogram A = lh

(f)     Perimeter of Square P = 4s

(g)   Perimeter of Rectangle P = 2L + 2W

(h)   Perimeter of Rhombus  P = 4s

(i)     Perimeter of Regular Polygon   P = ns

Answer. (a), (b), (c), (d), (e), (f), (h), (i)

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 the product of signs.

A positive (+) number multiplied by a positive (+) number yields a positive (+) product.
A positive (+) number multiplied by a negative (-) number yields a negative (-) product.
A negative (-) number multiplied by a positive (+) number yields a negative (-) product.
A negative (-) number multiplied by a negative (-) number yields a positive (+) product.

Then multiply the coefficients. Finally multiply the variables. To multiply the variables, use the Product Rule for Exponents.

For example, to multiply (–3xy²) by (–12xy²z²), follow the steps below. 

(a)   The product of the signs is positive. 

(b)   The product of the coefficients (3 and 12) is 36. 

(c)   Using the product Rule for Exponents, we get 

(x) × (x) = x² 

^{(Notice that when you are multiplying variables with no exponents, the exponents are assumed to be 1.)}

y² × y² = y⁴ 

(Notice that when you are multiplying variables and there is no other variable like the one you are multiplying, like the z² , you will just copy it.)

So, (–3xy²)(–12xy²z²) = 36x²y⁴z² 

Practice 3. Multiply. 

(a)   3x(xyz) 

(b)   (–21xyz)(3x²y³) 

(c)   (11mnx²)(–2mny²) 

(d)   (23xyz)(x²y³z⁴)

Solution.

(a)   3x(xyz) = 3(x^{1 + 1})(yz) 

            = 3x²yz 

(b)   (–21xyz)(3x²y³) = (–21 × 3)(x^{1 + 2})(y^{1 + 3})(z) 

         = –63x³y⁴z 

(c)   (11mnx²)(–2mny²) = (– 11 × 2)(m^{1 + 1})(n^{1 + 1})(x²y²) 

   = –22m²n²x²y² 

(d)   (23xyz)(x²y³z⁴) = (23)(x^{1 + 2})(y^{1 + 3})(z^{1 + 4}) 

         = 23x³y⁴z⁵ 

Practice 4. Determine the shapes whose areas or perimeters are in the form of monomial.

Answers. Rectangle:  A = lw = y(y + 4) = y²+ 4y
                                       P = 2l + 2w = 2(y + 4) + 2y = 2y + 8 + 2y = 4y + 8
                  Triangle: We cannot determine its area because the height is not given.
                                      P = 4x + 3x + 2x = 9x
                  Rhombus: We cannot determine its area because the height is not given.
                                     P = 4s = 4(ax) = 4ax
                                    Square: A = s²= 4(mp) = 4mp

The shape whose area is in the form of a monomial, is the square. The shapes whose perimeters are in the forms of monomials are the triangle, rhombus and square.

How to Multiply a Monomial by a Binomial?

To multiply a monomial by a binomial, use the Distributive Property.

For example, to multiply –2xy² by (x – 3y²) multiply the monomial by each term of the binomial.

–2xy²(x – 3y²) = –2xy²(x) –2xy²(–3y²)

                                              = –2x²y² + 6xy⁴

Also, you can use the special product identities below to multiply two binomials.

Practice 5. Multiply.

(a)   4x(2x + 3xy)

(b)   –12mn(m + n)

(c)   21m(mx² + mn²)

Solution.

(a)   4x(2x + 3xy) = 4x(2x) + 4x(3xy)

= 8x² + 12x²y

(b)   –12mn(m + n) = –12mn(m) – 12mn(n)

        = –12m²n –12mn²

(c)   21m(mx² + mn²) = 21m(mx²) + 21m(mn²)

= 21m²x² + 21m²n²

Practice 6. Use the identities to find each product.

(a)   (a + 2b)²

(b)   (m – 3n)²

(c)   (x + xy)(x – xy)

(d)   (1 + 2a)³

(e)   (3 – b)³

Solution.

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

         = a² + 4ab + 4b²

(b)   (m – 3n)² = m² – 2(m)(3n) + (3n)²

         = m² – 6mn + 9n²

(c)   (x + xy)(x – xy) = x² – (xy)²

        = x² – x²y²

(d)   (1 + 2a)³ = 1³ + 3(1²)(2a) + 3(1)(2a)² + (2a)³

         = 1 + 6a + 12a² + 8a³

(e)   (3 – b)³ = 3³ – 3(3²)(b) + 3(3)(b²) – b³

       = 27 – 27b + 9b² – b³ 

Geometric Application. Find the area of each shape.

Practical Exercise 1. Perform indicated operations.

(a)   –3x(x³y³)

(b)   12mn³(x³y)

(c)   (m + 2n)²

(d)   (2m – n)²

(e)   (a + ab)(a – ab)

(f)     (2m – 3n)³

(g)   (a + 2b)³

Answers