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

–4m³b + 2n³b² – a²b⁴ – ab

– x²y³z⁴ + 2ayz⁴ – 4xyz – x⁴y⁵z 

4mn³ – 3x³ – 4mn³ – 5y²x² – mny³z⁴

Opposite of a Polynomial

To find the opposite of a polynomial, simply reverse the sign of each term. For example, the opposite of –3x⁵ – 3x⁴ – 2x² – 6x – 5 is 3x⁵ + 3x⁴ + 2x² + 6x + 5 with all positive terms.

Practice 11. Find opposite of each polynomial.

(a)   31x⁵ – 3x³ + 25x² – 46x + 45

(b)   –56x⁸ – 3x⁷ + 29x⁵ + 65x³ – 43x

(c)   x⁷ + 33x⁵ + 24x⁴ – 61x² – 35

Solution.

(a)   –31x⁵ + 3x³ – 25x² + 46x – 45

(b)   56x⁸ + 3x⁷ – 29x⁵ – 65x³ + 43x

(c)   –x⁷ – 33x⁵ – 24x⁴ + 61x² + 35

Practice 12. Identify each expression as monomial, binomial, trinomial, or polynomial. 

(a)   3x + xyz 

(b)   11x²y³ 

(c)   321mnx² –2mny 

(d)   xyz – x²y³z⁴ + 23ay²z⁴ –43xyz – x²y³z⁶ 

(e)   3xyz – 3xz 

(f)     –22xy + z 3x² – y³ 

(g)   10mnx² –12m + ny² 

(h)   –3xyz 

(i)     21x + yz – 3x² + y³ x² + 32y – 3z⁴ 

(j)      mn – x² – 2mn – y²x² – y³z⁴

Answer.

(a)   Binomial
 

(b)   Monomial
 

(c)   Binomial
 

(d)   Polynomial
 

(e)   Binomial
 

(f)     Trinomial

(g)   Trinomial

(h)   Monomial

(i)     Polynomial

(j)      Polynomial

Practice 13. List geometric formulas in which a monomial is multiplied by a binomial.

Answer.

Perimeter of Rectangle = 2(a + b)

Perimeter of Parallelogram = 2(a + b)

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.

For example, if the first polynomial has 5 terms, we must multiply its terms one by one by the other polynomial. In this case, we will use the Distributive Property five times.

Example. Multiply

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

Multiply each term of the first polynomial by the other polynomial first.

(3x³ – 5x² – 2x +5)(x⁴ + 2x³ – x² + 7x – 2) = (3x³)(x⁴ + 2x³ – x² + 7x – 2)

+ (– 5x²)(x⁴ + 2x³ – x² + 7x – 2)

+ (–2x)(x⁴ + 2x³ – x² + 7x – 2)

+ (5)(x⁴ + 2x³ – x² + 7x – 2)

Use Distributive Property to find the product of monomials by polynomials.

(3x³ – 5x² – 2x + 5)(x⁴ + 2x³ – x² + 7x – 2) =

(3x³)(x⁴) + (3x³)(2x³) + (3x³)(– x²) + (3x³)(7x) +  (3x³)(– 2) =

(– 5x²)(x⁴) + (– 5x²)(2x³) + (–5x²)(– x²) + (–5x²)(7x) + (– 5x²)(– 2) +

(–2x)(x⁴) + (–2x)(2x³) + (–2x)(–x²) + (–2x)(7x) + (–2x)(–2) +

(5)(x⁴) + (5)(2x³) + (5)(–x²) + (5)(7x) + (5)(– 2)

Now, use the rule of multiplying a monomial by a monomial.

(3x³ – 5x² – 2x + 5)(x⁴ + 2x³ – x² + 7x – 2) =

3x⁷ + 6x⁶ – 3x⁵ + 21x⁴ – 6x³ – 5x⁶ – 10x⁵ + 5x⁴ – 35x³ + 10x²

–2x⁵ – 4x⁴ + 2x³ – 14x² + 4x + 5x⁴ + 10x³ – 5x² + 35x –10

Combine like terms.

(3x³ – 5x² – 2x + 5)(x⁴ + 2x³ – x² + 7x – 2) =

                 = 3x⁷ + x⁶ – 15x⁵ + 27x⁴ – 29x³ – 9x² + 39x– 10

Practice 14. Find a polynomial representing the area of the rectangle below.

Figure 6.4

Solution.

Area of Rectangle = (x + y)(x + y – z)

                                = x(x + y – z) + y(x + y – z)

        = x² + xy – xz + xy + y² – yz

                                = x² + 2xy – xz + y² – yz

Practical Exercise 4. Find each product.

(a)   (x² + x  + 1)(x⁴ + x³ – x² + 1)

(b)   (mn – m – n + 1)(m + n + 1)

Answer