7th Degree Polynomial Functions- Examples and Solutions

What Is a 7th Degree Polynomial Function?

A 7th degree polynomial function is a polynomial where the highest exponent of the variable is 7. The general form looks like this:

f(x) = ax⁷ + bx⁶ + cx⁵ + dx⁴ + ex³ + fx² + gx + h

Where a through h are constants, and a ≠ 0 (otherwise it's not actually a 7th degree polynomial).

The degree tells you key things:

• Maximum number of real roots: 7
• Maximum number of turning points: 6
• End behavior: Both ends go in opposite directions depending on the leading coefficient

Key Properties of 7th Degree Polynomials

Leading Coefficient Rules

If a > 0, the graph goes down on the left and up on the right (like a ↗️). If a < 0, it goes up on the left and down on the right (like a ↖️).

This is called odd-degree behavior. Unlike even-degree polynomials that both ends go the same direction, odd-degree polynomials always have opposite ends.

Roots and Factors

A 7th degree polynomial can have:

• 7 real roots (all crossing the x-axis)
• 5 real roots + 1 pair of complex conjugate roots
• 3 real roots + 2 pairs of complex conjugates
• 1 real root + 3 pairs of complex conjugates

Complex roots always come in pairs. That's non-negotiable.

7th Degree Polynomial Examples and Solutions

Example 1: Simple 7th Degree Polynomial

f(x) = x⁷ - x⁵

This factors nicely:

f(x) = x⁵(x² - 1) = x⁵(x - 1)(x + 1)

Roots are: x = 0 (with multiplicity 5), x = 1, x = -1

The root at x = 0 is a triple root visually even though it's multiplicity 5 — the graph touches and bounces at the origin. The other two roots cross the x-axis.

Example 2: Complete 7th Degree with Synthetic Division

f(x) = x⁷ - 4x⁶ - 7x⁵ + 10x⁴ + 28x³ - 16x² - 48x

Notice there's no constant term. That means x = 0 is a root. Let's factor out x first:

f(x) = x(x⁶ - 4x⁵ - 7x⁴ + 10x³ + 28x² - 16x - 48)

Now use synthetic division to find other factors. Testing x = 2:

The remainder is 0, so (x - 2) is a factor. The quotient is x⁶ - 2x⁵ - 11x⁴ - 12x³ + 4x² + 8x + 16.

Example 3: Finding Roots Given One Root

If you know x = 1 is a root of f(x) = x⁷ - 3x⁶ + 2x⁵ + x⁴ - x³ + 4x² - 3x + 2, find the remaining factors.

Use synthetic division with x = 1:

Quotient: x⁶ - 2x⁵ + 0x⁴ + x³ + 0x² + 4x - 3

Now you have a 6th degree polynomial to factor further. Test x = -1:

Evaluating at x = -1: (-1)⁶ - 2(-1)⁵ + 0 + (-1)³ + 0 + 4(-1) - 3 = 1 + 2 - 1 - 4 - 3 = -5 ≠ 0

Test x = 3:

3⁶ - 2(3⁵) + 0 + 3³ + 0 + 4(3) - 3 = 729 - 2(243) + 27 + 12 - 3 = 729 - 486 + 36 = 279 ≠ 0

Keep testing rational roots using the ± factors of constant ÷ factors of leading coefficient rule.

How to Work with 7th Degree Polynomials

Step 1: Check for Common Factors

Look for an x you can factor out. If every term has an x, factor it out first. This drops the degree by 1 and makes synthetic division easier.

Step 2: Find One Root

Test obvious candidates first:

• ±1 always worth trying
• Factors of the constant term
• Factors of the leading coefficient (for rational roots)

Step 3: Use Synthetic Division

Once you find a root r, divide by (x - r) to drop the degree. Repeat until you're working with a polynomial you can solve directly or graph.

Step 4: Determine Multiplicity

If synthetic division gives you a remainder of 0, you've found a root. If the quotient still has 0 as a root when you test again, that's multiplicity. A root with multiplicity k touches the x-axis but doesn't cross it.

Degree Comparison Table

Odd-degree polynomials always have opposite end behaviors. The degree number just determines how many wiggles fit between those ends.

Graphical Behavior You Need to Know

A 7th degree polynomial can have up to 6 turning points. That means the graph can change direction up to 6 times. With 7 roots, you're looking at a graph that crosses the x-axis up to 7 times.

In practice, most 7th degree polynomials you'll encounter in homework have been constructed to factor nicely. Look for:

• Opposite terms that cancel (like x⁷ - x⁵)
• Patterned coefficients
• Roots at ±1, ±2 that are easy to test

When You Need the Rational Root Theorem

If a 7th degree polynomial doesn't factor by grouping or obvious patterns, apply the Rational Root Theorem:

Any rational root p/q must have p dividing the constant term and q dividing the leading coefficient.

For f(x) = 2x⁷ - 3x⁶ + 4x⁵ - x⁴ + 6x³ - 9x² + 2x - 4, rational candidates are ±1, ±2, ±4, and ±1/2.

Start with the easy ones. Test until you find a root, then synthetic divide and repeat.

Bottom Line

7th degree polynomials are manageable if you:

• Factor out common x terms immediately
• Use synthetic division aggressively
• Test rational candidates systematically
• Track multiplicity when roots repeat

The math is straightforward. The arithmetic is where people mess up. Double-check your synthetic division — one wrong digit and everything downstream is wrong.