Algebra of Polynomials

Hello there! Great to continue our discussion on polynomials! You’re keen to understand the Algebra of Polynomials, specifically Addition and Subtraction, with ease and emojis. Let’s get right to it! ➕➖

What is a Polynomial? (A Quick Recap! 🔄)

Before we add and subtract, let’s briefly recall what a polynomial is. From a “Layman’s perspective,” a polynomial is a mathematical expression that is essentially a sum of several mathematical terms. Each of these “mathematical terms” can be a number, a variable, or a product of several variables. For instance, 3x is a term, x²y is a term, and 10 is a term.

From a “mathematician’s perspective,” a polynomial is an algebraic expression where the only arithmetic operations allowed are addition, subtraction, multiplication, and variables can only have “natural exponents”. “Natural exponents” mean non-negative integers (0, 1, 2, and so on). For example, t^(1/2) + t would not be a polynomial because t^(1/2) has a rational exponent, not a natural one.

The word “polynomial” itself comes from “poly” (meaning “many”) and “nomen” (meaning “terms”). Each individual term in a polynomial is called a monomial.

Addition of Polynomials 🤝

When you add two polynomials, the main idea is to combine “like terms”.

• What are “Like Terms”? 🤔 “Like terms” are terms that have the same variable and the same exponent. For example, 3x² and 5x² are like terms because they both have the variable x raised to the power of 2. However, 3x² and 5x³ are not like terms.

• The Process: 🧑‍💻

  1. Identify the terms in each polynomial.
  2. Align like terms. If a term is missing in one polynomial, you can think of it as having a coefficient of zero. For example, x² + 4 can be written as x² + 0x + 4 when adding it to a polynomial that has an x term.
  3. Add the coefficients of the like terms.

• General Formula (for polynomials in one variable): 📝 If you have two polynomials, p(x) and q(x): p(x) = a₀x⁰ + a₁x¹ + ... + aₙxⁿ q(x) = b₀x⁰ + b₁x¹ + ... + bₘxᵐ

  Their sum p(x) + q(x) is given by: p(x) + q(x) = ∑(a_k + b_k)x^k up to the maximum degree (m ∨ n, which means “whichever is maximum”).

• Resultant Degree: The degree of the polynomial you get after addition will be the maximum of the degrees of the two original polynomials. The term containing the highest degree will “survive”.

• Example: Let’s add p(x) = x³ + 3x² + 5x − 10 and q(x) = 3x³ + 5x² − 6x − 20: p(x) = 1x³ + 3x² + 5x¹ − 10x⁰ q(x) = 3x³ + 5x² − 6x¹ − 20x⁰

  Adding term by term (by combining coefficients of like terms):

  • For x³ terms: (1 + 3)x³ = 4x³
  • For x² terms: (3 + 5)x² = 8x²
  • For x¹ terms: (5 + (-6))x¹ = -1x¹
  • For x⁰ (constant) terms: (-10 + (-20))x⁰ = -30x⁰

  So, p(x) + q(x) = 4x³ + 8x² − x − 30. Notice the highest degree in both p(x) and q(x) was 3, and the resultant polynomial also has a degree of 3. 👍

Subtraction of Polynomials ➖

Subtracting polynomials is very similar to addition. The key difference is that you multiply the second polynomial by -1 before combining like terms.

• The Process: 🧑‍💻

  1. Change the sign of every term in the polynomial being subtracted. (This is equivalent to multiplying by -1).
  2. Combine the like terms as you would in addition.

• General Formula (for polynomials in one variable): 📝 If you have p(x) and q(x) as defined above: p(x) - q(x) = ∑(a_k - b_k)x^k up to the maximum degree (m ∨ n).

• Resultant Degree: The rules for determining the resultant degree in subtraction are the same as for addition. The degree of the result will be the maximum of the degrees of the original polynomials, unless the leading terms cancel out (e.g., subtracting 2x² from 2x² + 5).

• Example: Subtract q(x) = -3x² + 2x − 2 from p(x) = 1x³ + 2x² + 8x: First, rewrite p(x) and q(x) with all terms explicitly, including zero coefficients for missing terms if needed: p(x) = 1x³ + 2x² + 8x + 0 q(x) = 0x³ - 3x² + 2x - 2

  Now, “multiply” q(x) by -1 (change the sign of each term): -q(x) = 0x³ + 3x² - 2x + 2

  Now, add p(x) and -q(x):

  • For x³ terms: (1 + 0)x³ = 1x³
  • For x² terms: (2 + 3)x² = 5x²
  • For x¹ terms: (8 + (-2))x¹ = 6x¹
  • For x⁰ (constant) terms: (0 + 2)x⁰ = 2x⁰

  So, p(x) - q(x) = x³ + 5x² + 6x + 2. Here, the highest degree was 3, and the result is also degree 3. 👍

Practice Questions! 📝

Question 1: Add the following polynomials: P(x) = 7x⁴ + 3x² - 5 Q(x) = 2x⁴ - x³ + 4x - 1

Question 2: Subtract R(x) = -2x⁵ + 5x² + x - 10 from S(x) = x⁵ + 3x⁴ - 4x² + 7.

Question 3: What is the degree of the resultant polynomial when (4x² + 2x) is subtracted from (4x² + 5)?

Solutions to Practice Questions ✅

Solution 1: Add P(x) = 7x⁴ + 3x² - 5 and Q(x) = 2x⁴ - x³ + 4x - 1

Let’s align the terms and add their coefficients, treating missing terms as having zero coefficients:

P(x) = 7x⁴ + 0x³ + 3x² + 0x¹ - 5x⁰ Q(x) = 2x⁴ - 1x³ + 0x² + 4x¹ - 1x⁰

P(x) + Q(x) = (7+2)x⁴ + (0-1)x³ + (3+0)x² + (0+4)x¹ + (-5-1)x⁰ P(x) + Q(x) = 9x⁴ - x³ + 3x² + 4x - 6 🎉

The highest degree of P(x) is 4, and the highest degree of Q(x) is 4. The resultant polynomial has a degree of 4, which is the maximum of the two.

Solution 2: Subtract R(x) = -2x⁵ + 5x² + x - 10 from S(x) = x⁵ + 3x⁴ - 4x² + 7

First, align the terms of S(x) and R(x), filling in missing terms with zero coefficients: S(x) = 1x⁵ + 3x⁴ + 0x³ - 4x² + 0x¹ + 7x⁰ R(x) = -2x⁵ + 0x⁴ + 0x³ + 5x² + 1x¹ - 10x⁰

Now, change the sign of each term in R(x) (effectively multiplying R(x) by -1): -R(x) = +2x⁵ + 0x⁴ + 0x³ - 5x² - 1x¹ + 10x⁰

Now, add S(x) and -R(x): S(x) = 1x⁵ + 3x⁴ + 0x³ - 4x² + 0x¹ + 7x⁰ -R(x) = 2x⁵ + 0x⁴ + 0x³ - 5x² - 1x¹ + 10x⁰

S(x) - R(x) = (1+2)x⁵ + (3+0)x⁴ + (0+0)x³ + (-4-5)x² + (0-1)x¹ + (7+10)x⁰ S(x) - R(x) = 3x⁵ + 3x⁴ + 0x³ - 9x² - x + 17 S(x) - R(x) = 3x⁵ + 3x⁴ - 9x² - x + 17 🥳

The highest degree of S(x) is 5, and the highest degree of R(x) is 5. The resultant polynomial has a degree of 5, which is the maximum of the two.

Solution 3: What is the degree of the resultant polynomial when (4x² + 2x) is subtracted from (4x² + 5)?

Let P(x) = 4x² + 5 and Q(x) = 4x² + 2x. We want to find the degree of P(x) - Q(x).

P(x) = 4x² + 0x + 5 Q(x) = 4x² + 2x + 0

Change the signs of Q(x): -Q(x) = -4x² - 2x - 0

Now add P(x) and -Q(x): P(x) - Q(x) = (4 - 4)x² + (0 - 2)x + (5 - 0) P(x) - Q(x) = 0x² - 2x + 5 P(x) - Q(x) = -2x + 5

The highest power of x in the resultant polynomial is x¹. Therefore, the degree of the resultant polynomial is 1. 📉

This shows a case where the “maximum degree” rule for addition/subtraction (which typically assumes the leading coefficients don’t cancel) still applies in principle, but the effective degree can be lower if the highest-degree terms cancel out.