Polynomials: Expressions with Multiple Powers of x

A polynomial is a sum of terms, each with a whole number power of x.

That's it. x² + 3x + 5 is a polynomial. So is x⁷ - 2x³ + x. So is just "7" (a constant polynomial).

Here's the unlock: polynomials are the simplest expressions you can build from addition, subtraction, multiplication, and non-negative integer powers. No division by x. No square roots of x. No x in exponents. Just sums of x-to-integer-powers times coefficients.

This simplicity makes polynomials the workhorses of algebra — they're easy to add, multiply, factor, and evaluate, and they can approximate almost any function you care about.

The Structure

A polynomial looks like:

aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀

Each aₖ is a coefficient (a number). Each xᵏ is a power of the variable.

Example: 3x⁴ - 2x² + 5x - 7

Terms: 3x⁴, -2x², 5x, -7 Coefficients: 3, -2, 5, -7 Powers: 4, 2, 1, 0

Degree

The degree of a polynomial is its highest power.

• 3x⁴ - 2x² + 5 has degree 4
• x² + x + 1 has degree 2
• 5x - 3 has degree 1
• 7 has degree 0 (constant)

The degree tells you the polynomial's "complexity" — how it grows as x gets large, and roughly how many solutions the equation polynomial = 0 can have.

Naming by Degree

Polynomials have names based on degree:

• Degree 0: Constant (just a number)
• Degree 1: Linear (a line: ax + b)
• Degree 2: Quadratic (parabola: ax² + bx + c)
• Degree 3: Cubic (S-curve: ax³ + ...)
• Degree 4: Quartic
• Degree 5: Quintic

You've already studied linear and quadratic. Cubics and higher follow similar patterns but with more complexity.

Adding and Subtracting Polynomials

Combine like terms — terms with the same power.

(3x² + 2x + 1) + (x² - 5x + 4) = (3x² + x²) + (2x - 5x) + (1 + 4) = 4x² - 3x + 5

(5x³ - 2x) - (3x³ + x² - 2x) = 5x³ - 2x - 3x³ - x² + 2x = 2x³ - x²

Group by power, then add coefficients.

Multiplying Polynomials

Multiply each term in the first by each term in the second, then combine like terms.

(x + 2)(x + 3) = x·x + x·3 + 2·x + 2·3 = x² + 3x + 2x + 6 = x² + 5x + 6

This is FOIL for binomials: First, Outer, Inner, Last.

For longer polynomials, same idea — just more terms.

(x + 1)(x² + 2x + 3) = x(x² + 2x + 3) + 1(x² + 2x + 3) = x³ + 2x² + 3x + x² + 2x + 3 = x³ + 3x² + 5x + 3

The Degree of Products

When you multiply polynomials, the degrees add.

(degree 2) × (degree 3) = degree 5

Why? The highest power in the product comes from multiplying the highest powers in each factor.

(x²)(x³) = x⁵

Special Products

Square of a binomial: (a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²

Difference of squares: (a + b)(a - b) = a² - b²

Cube of a binomial: (a + b)³ = a³ + 3a²b + 3ab² + b³

These patterns appear constantly. Recognizing them speeds up both multiplication and factoring.

Polynomial Division

You can divide polynomials using long division, similar to dividing numbers.

(x² + 5x + 6) ÷ (x + 2)

Divide leading terms: x² ÷ x = x Multiply: x(x + 2) = x² + 2x Subtract: (x² + 5x + 6) - (x² + 2x) = 3x + 6 Repeat: 3x ÷ x = 3 Multiply: 3(x + 2) = 3x + 6 Subtract: 0

Result: x + 3 with no remainder.

Check: (x + 2)(x + 3) = x² + 5x + 6 ✓

The Remainder Theorem

If you divide polynomial P(x) by (x - a), the remainder is P(a).

Example: Divide x² + 5x + 6 by (x - 1).

The remainder equals P(1) = 1 + 5 + 6 = 12.

This connects polynomial division to evaluation — useful for checking factors.

The Factor Theorem

(x - a) is a factor of P(x) if and only if P(a) = 0.

If P(2) = 0, then (x - 2) is a factor.

Example: P(x) = x² - 5x + 6

P(2) = 4 - 10 + 6 = 0, so (x - 2) is a factor. P(3) = 9 - 15 + 6 = 0, so (x - 3) is a factor.

Indeed: x² - 5x + 6 = (x - 2)(x - 3)

Zeros and Roots

A zero (or root) of a polynomial is a value where P(x) = 0.

The Fundamental Theorem of Algebra says a polynomial of degree n has exactly n zeros (counting multiplicity and complex numbers).

• Linear (degree 1): 1 zero
• Quadratic (degree 2): 2 zeros
• Cubic (degree 3): 3 zeros

Not all zeros are real. x² + 1 = 0 has no real solutions — its zeros are complex (±i).

End Behavior

As x → ∞ or x → -∞, a polynomial is dominated by its highest-degree term.

For P(x) = 3x⁴ - 2x² + 5:

• As x → ∞, P(x) → ∞ (positive leading coefficient, even degree)
• As x → -∞, P(x) → ∞ (even degree means both ends go same direction)

For P(x) = -2x³ + x + 1:

• As x → ∞, P(x) → -∞ (negative leading coefficient)
• As x → -∞, P(x) → ∞ (odd degree means opposite ends)

The degree and leading coefficient determine the overall shape.

Why Polynomials Are Special

Polynomials are closed under addition, subtraction, and multiplication. Add two polynomials, get a polynomial. Multiply two polynomials, get a polynomial.

They're also easy to evaluate: just plug in a number and compute.

And they can approximate almost any smooth function (Taylor series). This makes them fundamental to calculus, physics, and engineering.

Not Polynomials

These are NOT polynomials:

• 1/x (negative power: x⁻¹)
• √x (fractional power: x^(1/2))
• 2ˣ (variable in the exponent)
• sin(x) (transcendental function)

Polynomials have only non-negative integer powers of x with constant coefficients.

The Core Insight

A polynomial is a sum of x-power terms.

The degree tells you its complexity: how many zeros it can have, how it grows, what its graph looks like.

All the operations you know — adding, multiplying, factoring — work on polynomials in predictable ways because the structure is constrained. No fractions, no roots, no weird functions. Just x raised to integer powers, added together.

Polynomials are algebra's bread and butter. Master them, and the rest builds on top.

Part 9 of the Algebra Fundamentals series.

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