Polynomials Unit Test- Lesson 11 Assessment Review

What This Review Actually Covers

Lesson 11 wraps up your polynomials unit. The test will hit you on three main areas: operations, factoring, and graphing. That's it. Everything else is noise.

If you can't add, subtract, multiply, divide, factor, or graph a polynomial without hesitating, you will fail. There's no curve that saves you here.

The Basics You Better Know Cold

A polynomial is a sum of terms with variables raised to whole number exponents. No fractions. No negative exponents. No radicals hiding in there.

Each term has three parts you need to track:

• The coefficient (the number in front)
• The variable base (usually x)
• The exponent (the power)

The degree of a polynomial is just the highest exponent. That's it. Don't overthink it.

Classifying by Degree

This comes up more than you'd think:

• Degree 0 = constant
• Degree 1 = linear
• Degree 2 = quadratic
• Degree 3 = cubic
• Degree 4 = quartic

Operations With Polynomials

Adding and Subtracting

Combine like terms only. Like terms have the same variable part. Everything else stays separate.

Example: 3x² + 2x - 5 + 4x² - 3x + 1

Combine: (3x² + 4x²) + (2x - 3x) + (-5 + 1) = 7x² - x - 4

If this trips you up, you're not ready for the test.

Multiplying Polynomials

Distribute everything to everything. That's the rule. No exceptions.

For a monomial times a polynomial: multiply the monomial by each term.

For binomial times binomial: use FOIL. First, Outer, Inner, Last. Yes, it's a real method. No, it's not magic. It's just distribution with a better name.

For anything larger: distribute each term in the first polynomial to every term in the second. Then combine like terms.

Dividing Polynomials

You'll use long division or synthetic division. Synthetic is faster but only works when dividing by a linear expression in the form (x - c).

For long division: divide, multiply, subtract, bring down. Repeat until done. If you forget a term while writing the problem, add a zero placeholder. Forgetting this costs you points.

Factoring Polynomials

Factoring is where most students fall apart. Here's the hierarchy:

Step 1: Check for GCF First

Always. Look for the greatest common factor in every term. Factor it out before doing anything else.

Example: 12x³ + 18x² = 6x²(2x + 3)

Step 2: Count the Terms

• Two terms: Look for difference of squares (a² - b²), sum of cubes, or difference of cubes
• Three terms: Try trial and error with factors, or use the quadratic formula to find roots first
• Four terms: Try grouping — factor the first two, factor the last two, then factor out what's common

Special Products to Memorize

These show up constantly. If you don't have them memorized, the test will eat you alive.

Graphing Polynomials

You need to know these key features:

• End behavior: What happens as x goes to +∞ and -∞. Look at the leading coefficient and degree.
• Y-intercept: Set x = 0 and solve. That's your y-intercept.
• X-intercepts: These are the roots. Factor the polynomial to find them.
• Multiplicity: If a root appears multiple times, it affects how the graph touches or crosses the x-axis. Even multiplicity = touches, odd multiplicity = crosses.

The Zeros Test

To find zeros: set the polynomial equal to zero and solve. Every factor (x - r) gives you a zero at x = r. Count how many times each factor appears — that's the multiplicity.

Solving Polynomial Equations

Set the polynomial equal to zero. Factor it. Apply the zero product property: if AB = 0, then A = 0 or B = 0.

Check your answers. Plug them back into the original equation. Reject anything that makes the equation undefined.

Common Mistakes That Cost Points

• Dropping negative signs during distribution — write each step if you have to
• Forgetting to include all terms when multiplying binomials
• Not checking for a GCF before attempting other factoring methods
• Losing track of terms in long division
• Confusing the degree of a polynomial with the number of terms
• Forgetting that factoring is the reverse of distribution — check your work by multiplying back

Getting Started: Your Study Plan

Step 1: Sort your notes by topic. Group operations together, factoring together, graphing together.

Step 2: Work through five problems from each category. No looking at answers first.

Step 3: Check your answers. If you got it wrong, figure out why before moving on.

Step 4: Redo the problems you missed without help. If you can't, you don't understand it yet.

Step 5: Time yourself. The test has a limit. If you're spending five minutes on one problem, something is wrong with your approach.

What to Bring to the Test

• Graphing calculator — know how to use it, not just that you have one
• Pencil and eraser — scratch paper fills up fast
• Your formula sheet — if you're allowed one, fill it out before the test starts

Don't waste time deriving formulas during the test. Memorize the special products. Memorize FOIL. Memorize the steps for synthetic division.

The Bottom Line

This test measures whether you can work with polynomials or not. There's no secret trick. There's no partial credit for effort.

Factor cleanly. Distribute completely. Check your work. That's the whole game.