Last modified on November 21st, 2024

chapter outline

 

Simplifying Polynomials

Simplifying polynomials means reducing the polynomials to their simplest form by following certain steps. 

Here are the general steps we follow to simplify polynomial expressions:

  1. Combining Like Terms
  2. Expanding and Simplifying Terms 
  3. Rearranging Terms into Standard Form

Let us simplify the polynomial 3x2 + 2x – x2 + 4x + x2(4x + 3) + (x + 3)(x + 1) 

Combining Like Terms

Like terms are terms that have the same variable raised to the same exponent. The first step in simplifying a polynomial is to identify and combine like terms (the part marked bold) without expanding the other part of the expression. 

In the polynomial 3x2 + 2x – x2 + 4x + x2(4x + 3) + (x + 3)(x + 1), we have two types of like terms:

  • 3x2 and -x2 (both have the variable x raised to the exponent 2)
  • 2x and 4x (both have the variable x raised to the exponent 1)

By combining these like terms, we can simplify the polynomial as follows:

3x2 + 2x – x2 + 4x + x2(4x + 3) + (x + 3)(x + 1)

= (3x2 – x2) + (2x + 4x) + x2(4x + 3) + (x + 3)(x + 1)

= 2x2 + 6x + x2(4x + 3) + (x + 3)(x + 1) …..(i)

Expanding and Simplifying 

Now, we expand any remaining terms using the distributive property and use the FOIL method only for multiplying binomials to ensure all terms are fully simplified.

Expanding Using the Distributive Property

The distributive property states that a(b + c) = ab + ac

In the polynomial (i), we expand x2(4x + 3) using the distributive property:

x2(4x + 3) = x2 ⋅ 4x + x2 ⋅ 3 = 4x3 + 3x2 …..(ii)

Multiplying Using the FOIL Method

‘FOIL’ stands for Firsts, Outers, Inners, and Lasts. 

In the polynomial (i), we need to expand (x + 3)(x + 1) using the FOIL method:

Firsts: x ⋅ x = x2 

Outers: x ⋅ 1 = x

Inners: 3 ⋅ x = 3x

Lasts: 3 ⋅ 1 = 3

Thus, (x + 3)(x + 1) = x2 + x + 3x + 3 = x2 + 4x + 3 …..(iii)

Now, substituting (ii) and (iii) in the polynomial (i), we get

2x2 + 6x + x2(4x + 3) + (x + 3)(x + 1)

= 2x2 + 6x + 4x3 + 3x2 + x2 + 4x + 3

Further combining the remaining like terms, 

= (2x2 + 3x2 + x2) + (6x + 4x) + 4x3 + 3

= 6x2 + 10x+ 4x3 + 3

Rearranging in Standard Form

Finally, we arrange the polynomial in descending order of powers.

Thus, the polynomial is now in its simplified form:

3x2 + 2x – x2 + 4x + x2(4x + 3) + (x + 3)(x + 1) = 4x3 + 6x2 + 10x + 3

Solved Examples

Simplify the following polynomials:
a) 2x2 + 3x – (4x2 – 5x + 6) + x(3x + 2) – (x + 1)(x – 4)
b) 5y3 + 2y2 – (3y3 + y2 – y + 4) + (y + 2)(y – 3) + 6y

Solution:

a) Given, 2x2 + 3x – (4x2 – 5x + 6) + x(3x + 2) – (x + 1)(x – 4)
= 2x2 + 3x – 4x2 + 5x – 6 + 3x2 + 2x – (x2 – 4x + x – 4)
= 2x2 + 3x – 4x2 + 5x – 6 + 3x2 + 2x – x2 + 4x – x + 4
= (2x2 – 4x2 + 3x2 – x2) + (3x + 5x + 2x + 4x – x) + (-6 + 4)
= 13x – 2
b) Given, 5y3 + 3y2 – (3y3 + y2 – y + 4) + (y + 2)(y – 3) + 6y
= 5y3 + 3y2 – 3y3 – y2 + y – 4 + (y2 – 3y + 2y – 6) + 6y
= 5y3 + 3y2 – 3y3 – y2 + y – 4 + y2 – 3y + 2y – 6 + 6y
= (5y3 – 3y3) + (3y2 – y2 + y2) + (y – 3y + 2y + 6y) + (-4 – 6)
= 2y3 + 3y2 + 6y – 10

Simplify:
a) ${\dfrac{-8x^{-3}y^{-2}z^{4}}{2x^{-5}y^{3}z^{-2}}}$
b) (2x2 + 3x – 5) – (x2 + x – 9)
c) (7m)(3m2n)(2mn2) + (3m2)(4n3)(m2n)

Solution:

a) Given, ${\dfrac{-8x^{-3}y^{-2}z^{4}}{2x^{-5}y^{3}z^{-2}}}$
= ${\left( \dfrac{-8}{2}\right) \left( \dfrac{x^{-3}}{x^{-5}}\right) \left( \dfrac{y^{-2}}{y^{3}}\right) \left( \dfrac{z^{4}}{z^{-2}}\right)}$
= -4x-3 – (-5)y-2 – 3z4 – (-2) 
= -4x2y-5z6 
b) Given, (2x2 + 3x – 5) – (x2 + x – 9)
= 2x2 + 3x – 5 – x2 – x + 9
= (2x2 – x2) + (3x – x) + (-5 + 9)
= x2 + 2x + 4
c) Given, (7m)(3m2n)(2mn2) + (3m2)(4n2)(m2n)
= (7 ⋅ 3 ⋅ 2)(m ⋅ m2 ⋅ m)(n ⋅ n2) + (3 ⋅ 4)(m2 ⋅ m2)(n2 ⋅ n)
= 42m4n3 + 12m4n3 = 54m4n3

Last modified on November 21st, 2024