9. Adding and Subtracting Polynomials

Notes:

Definition and terminology of polynomials

The algebraic expression, axⁿ + aₙ₋₁xⁿ⁻¹ + … + a₂x² + a₁x + a₀, represents a polynomial of degree n in the variable x. Here n is a non-negative integer.

• The a’s, called coefficients, are real number constants with leading coefficient aₙ ≠ 0.
• Each product of a coefficient and a power of x is called a term with leading term aₙxⁿ and constant term a₀.

For example, 2x³ – 4x² + 1:

• Degree: 3.
• Leading term: 2x³ with leading coefficient 2.
• Coefficient of x² is -4.
• Coefficient of x is 0.
• Constant term is 1.

Adding and subtracting polynomials

To add and subtract polynomials, we must compare terms with the same power of the variable. For example, to add (2x³ – 4x² + 1) and (3x³ + 5x² + 2x – 4):

• Add 2x³ and 3x³ to get (2 + 3)x³ = 5x³.
• Add -4x² and 5x² to get (-4 + 5)x² = x².
• Add 0x and 2x to get (0 + 2)x = 2x.
• Add 1 and -4 to get 1 + (-4) = -3.
• The final answer is therefore 5x³ + x² + 2x – 3.

Similarly, (2x³ – 4x² + 1) – (3x³ + 5x² + 2x – 4) = -x³ – 9x² – 2x + 5.

Multiplying a polynomial by a constant

Think of multiplying a polynomial by a constant as repeated addition or subtraction. For example:

• 2(2x³ – 4x² + 1) = (2 × 2x³) + (2 × -4x²) + (2 × 1) = 4x³ – 8x² + 2.
• -3(2x³ – 4x² + 1) = -6x³ + 12x² – 3.

More examples:

• (2x³ – 5x² + 4) + (3x² + 2x – 4) = 2x³ – 3x² + 2x.
• (x² + 3x – 7) – (3x² – 2x + 1) = -2x² + 5x – 8.
• (2x² + 3xy + 4x) + (4y² – 3xy – 2x + 2) = 2x² + 4y² + xy + 2x + 2.

Adding and subtracting polynomials with two variables:

• (2x² + xy + 2y + 1) – (x² – 2y² + 3x – 2y) = x² + xy + 2y² – x + 2y + 1.
• (x² + 3xy + 4x + 2) + (4y² – 3xy – 2x + 2) = x² + 4y² + 2x + 2.