Polynomials

Updated: Jun 22, 2021



In this note, you will learn:


· Multiplication of polynomials

· Equality of polynomials

· Division of polynomials

Multiplication of polynomials

Back in lower secondary math, we have learnt that the product of algebraic expressions can be found using the Distributive Law.




Similarly, the same law can be applied when we are trying to find the product of polynomials.


Let’s take a look at an example below:


If P(x) = 3x^5 – 6x^4 and Q(x) = 4x^2 + 8x – 7, find


i) P(x) x Q(x)

ii) the relationship between the degrees of P(x), Q(x) and P(x) x Q(x).


Solution:

i) P(x) x Q(x) = (3x^5 – 6x^4) (4x^2 + 8x – 7)

= 12x^7 + 24x^6 – 21x^5 – 24x^6 – 48x^5 + 42x^4

= 12x^7 – 69x^5 + 42x^4


ii) Degree of P(x) = 5

Degree of Q(x) = 2

Degree of P(x) x Q(x) = 7


Therefore, degree of P(x) x Q(x) = degree of P(x) + degree of Q(x)



Equality of polynomials


In the scenario where two polynomials are given, P(x) = Ax^4 + Bx^3 + Cx^2 + Dx + E and Q(x) = 5x^4 + 2x^3 – 3x^2 + 8x – 15, and they are said to be equivalent, then the equation P(x) = Q(x) will be true for all real values of x. i.e. A = 5, B = 2, C = -3, D = 8 and E = -15. We call the equation a polynomial identity.



Let’s take a look at an example of how this works below:


Given that 6x^2 – 8x + 9 = a (x – 1) (x + 2) + b (x – 4) + c for all values of x, find the values of a, b and c.



Solution:

There are two methods for this type of questions: Substitution and Equating the coefficient.


Method 1: Substitution


Let x = 1: 6(1)^2 – 8(1) + 9 = a [(1) – 1] [(1) + 2] + b [(1) – 4] + c

6 – 8 + 9 = -3b + c

c = 7 + 3b -> Equation 1


Let x = -2: 6(-2)^2 – 8(-2) + 9 = a [(-2) – 1] [(-2) + 2] + b [(-2) – 4] + c

24 + 16 + 9 = -6b + c

49 = -6b + c -> Equation 2


Substituting Equation 1 into Equation 2,

49 = -6b + (7 + 3b)

42 = -3b

Therefore, b = -14


Substituting b = -14 into Equation 1,

c = 7 + 3(-14)

c = 7 – 42

Therefore, c = -35


Let x = 4: 6(4)^2 – 8(4) + 9 = a [(4) – 1] [(4) + 2] + (-14) [(4) – 4] + (-35)

96 – 32 + 9 = a (3) (6) - 35 a = 6



Therefore, a = 6, b = -14 and c = -35


Note: The values we let x to be is not completely random! It is wise to pick values of x that eliminates a certain term so that only one or two unknown coefficients are left, which we can then easily work them out using substitution. i.e. Given that we have a (x – 3) + b (x + 4), we will let x = 3 to eliminate the term with coefficient a first so that we can find coefficient b.



Method 2: Equating coefficients


6x^2 – 8x + 9 = a (x – 1) (x + 2) + b (x – 4) + c

= a (x^2 + x – 2) + b (x – 4) + c

= ax^2 + (a + b) x – 2a – 4b + c


Equating coefficients of x^2: 6 = a


Equating coefficients of x: -8 = a + b

-8 = (6) + b

b = -14


Equating coefficients of x^0: 9 = -2a – 4b + c

9 = -2(6) – 4(-14) + c