Polynomials and Factoring

Polynomials & Factoring

Part 1 of 7 — Polynomial Basics

What is a Polynomial?

A polynomial is an expression with one or more terms: $a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$

• Degree: highest power of $x$ (e.g., $3x^4 + 2x - 1$ has degree 4)
• Leading coefficient: coefficient of the highest-degree term
• Constant term: the term with no variable ($a_0$)

Adding & Subtracting Polynomials

Combine like terms (same variable and exponent):

$(3x^2 + 5x - 2) + (x^2 - 3x + 7) = 4x^2 + 2x + 5$

$(3x^2 + 5x - 2) - (x^2 - 3x + 7) = 3x^2 + 5x - 2 - x^2 + 3x - 7 = 2x^2 + 8x - 9$

Subtraction trap: distribute the negative sign to ALL terms in the second polynomial!

Multiplying Polynomials

Use distribution (FOIL for binomials):

$(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$

Worked Example 1 — Multiplying Three Factors

Expand $(x + 1)(x - 3)(x + 2)$.

Worked Example 2 — Subtraction Trap

Simplify $(5x^3 - x^2 + 4) - (3x^3 + 2x^2 - x + 1)$.

Polynomial Operations 🎯

Special Products to Memorize

Worked Example 3 — Using Special Products

Expand $(3x - 2)^2$.

⚠️ Common mistake: $(3x - 2)^2 \neq 9x^2 - 4$. You must include the middle term!

Degree of a Product

The degree of a product equals the sum of the degrees.

$(2x^3 + 1)(x^2 - 5x) \to \text{degree } 3 + 2 = 5$

Special Products & Degree 🎯

Identify the Property 🔍

For each expression, identify what type of operation or pattern it represents.

Key Takeaways — Part 1

• Combine like terms as the final step after every operation
• FOIL is just distribution for two binomials — use full distribution for larger products
• Special products save time and reduce errors on the SAT

Polynomials & Factoring

Part 2 of 7 — Factoring Techniques

GCF Factoring

Always look for a Greatest Common Factor first:

$6x^3 + 9x^2 = 3x^2(2x + 3)$

Difference of Squares

$a^2 - b^2 = (a + b)(a - b)$

Example: $x^2 - 49 = (x + 7)(x - 7)$

Tricky example: $4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5)$

Perfect Square Trinomials

$a^2 + 2ab + b^2 = (a + b)^2$ $a^2 - 2ab + b^2 = (a - b)^2$

How to recognize: first and last terms are perfect squares, middle term is $\pm 2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$.

$x^2 + 10x + 25 = (x + 5)^2$ because $2(x)(5) = 10x$ ✓

Sum/Difference of Cubes (Rare on SAT)

$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Worked Example 1 — Multi-Step Factoring

Factor completely: $2x^3 - 18x$.

Worked Example 2 — Recognizing Perfect Squares

Is $9x^2 - 30x + 25$ a perfect square trinomial?

Factoring Patterns 🎯

Factoring Trinomials: $ax^2 + bx + c$

When $a = 1$: Find two numbers that multiply to $c$ and add to $b$.

$x^2 + 7x + 12$: numbers that multiply to $12$ and add to $7$? → $3$ and $4$.

$= (x + 3)(x + 4)$

When $a \neq 1$: Use the AC method.

Worked Example 3 — AC Method

Factor $6x^2 + 11x - 10$.

Worked Example 4 — Signs Guide

Trinomial Factoring 🎯

What Factoring Method? 🔍

Choose the best factoring technique for each expression.

Key Takeaways — Part 2

• "Factor completely" = keep going until nothing else factors
• $a^2 + b^2$ does NOT factor over the reals
• Signs of $b$ and $c$ tell you the signs of the factor pair

Polynomials & Factoring

Part 3 of 7 — Polynomial Division

Long Division

Divide $x^3 + 2x^2 - 5x + 6$ by $x - 1$:

1. $x^3 \div x = x^2$. Multiply: $x^2(x-1) = x^3 - x^2$. Subtract: $3x^2 - 5x$
2. $3x^2 \div x = 3x$. Multiply: $3x(x-1) = 3x^2 - 3x$. Subtract: $-2x + 6$
3. $-2x \div x = -2$. Multiply: $-2(x-1) = -2x + 2$. Subtract: $4$

Result: $x^2 + 3x - 2$ remainder $4$.

Synthetic Division (Faster!)

For dividing by $(x - c)$: write the coefficients, bring down, multiply, add.

Dividing $x^3 + 2x^2 - 5x + 6$ by $(x - 1)$:

Result: $x^2 + 3x - 2$ with remainder $4$.

The Remainder Theorem

The remainder when $f(x)$ is divided by $(x - c)$ equals $f(c)$.

Check: $f(1) = 1 + 2 - 5 + 6 = 4$ ✓

Worked Example 1 — Missing Term Trap

Divide $x^3 - 8$ by $(x - 2)$.

So $x^3 - 8 = (x - 2)(x^2 + 2x + 4)$ — this is the difference of cubes formula!

Worked Example 2 — Using Remainder Theorem Strategically

Is $(x + 2)$ a factor of $2x^3 + x^2 - 7x + 2$?

Polynomial Division 🎯

Writing the Result of Division

$\frac{f(x)}{d(x)} = q(x) + \frac{r}{d(x)}$

Example: $\frac{x^2 + 3x + 5}{x + 1} = x + 2 + \frac{3}{x + 1}$

This form appears on the SAT! They may ask "what is the remainder" or "rewrite the expression."

Worked Example 3

Rewrite $\frac{x^2 - 4x + 7}{x - 1}$ in quotient-remainder form.

SAT Shortcut: Remainder without Division

To find just the remainder of $f(x) \div (x - c)$, simply compute $f(c)$. No division needed!

Division Applications 🎯

Remainder Theorem Quick Check 🔍

Find the remainder without doing long division.

Key Takeaways — Part 3

• Remainder Theorem: $f(x) \div (x-c)$ → remainder $= f(c)$
• Factor Theorem: $(x - c)$ is a factor iff $f(c) = 0$
• Include $0$ coefficients for missing terms (e.g., $x^3 - 8$ → $1, 0, 0, -8$)
• Division result: $f(x) = d(x) \cdot q(x) + r$

Polynomials & Factoring

Part 4 of 7 — Zeros, Roots, and the Factor Theorem

Zeros = Roots = x-intercepts

These terms all mean the same thing: the values of $x$ where $f(x) = 0$.

If $f(x) = (x - 2)(x + 5)(x - 1)$, the zeros are $x = 2, -5, 1$.

Multiplicity

The multiplicity of a zero is how many times its factor appears.

$f(x) = (x - 3)^2(x + 1)$:

• $x = 3$ has multiplicity 2 (graph touches x-axis and bounces)
• $x = -1$ has multiplicity 1 (graph crosses x-axis)

End Behavior

SAT Connection

The SAT asks: "How many x-intercepts does the graph of $f(x) = x^3 - 4x$ have?"

Factor: $x(x^2 - 4) = x(x-2)(x+2)$. Three distinct factors → 3 x-intercepts.

Worked Example 1 — Building a Polynomial from Zeros

Find a polynomial with zeros at $x = -1, 2, 5$ and leading coefficient $3$.

Worked Example 2 — Finding $a$ from a Point

$f(x) = a(x - 1)(x + 3)$ passes through $(2, 10)$. Find $a$.

Zeros & End Behavior 🎯

Multiplicity and Graph Behavior

Worked Example 3 — Degree from Graph

A graph crosses the x-axis at $x = -3$ and $x = 4$, and bounces at $x = 1$. Both ends point downward. What is the minimum degree?

Both ends down → even degree, negative leading coefficient ✓ (degree 4 is even).

Worked Example 4 — Number of Real Zeros

$f(x) = x^4 - 5x^2 + 4$. How many x-intercepts?

Graph Analysis 🎯

Match the Graph Feature 🔍

What does each piece of information tell you?

Key Takeaways — Part 4

• Given zeros + one point → find $a$ by substitution
• Number of real zeros ≤ degree of polynomial
• The SAT often shows a graph and asks for the equation — read zeros + end behavior first

Polynomials & Factoring

Part 5 of 7 — Rational Expressions

Simplifying Rational Expressions

A rational expression is a fraction with polynomials:

$\frac{x^2 - 9}{x^2 + 5x + 6} = \frac{(x+3)(x-3)}{(x+2)(x+3)} = \frac{x - 3}{x + 2} \quad (x \neq -3)$

Steps: Factor numerator and denominator, then cancel common factors.

Multiplying & Dividing

Multiply: Factor, cancel, then multiply what remains.

$\frac{x^2 - 4}{x + 1} \cdot \frac{x + 1}{x - 2} = \frac{(x+2)(x-2)}{x+1} \cdot \frac{x+1}{x-2} = x + 2$

Divide: Flip the second fraction and multiply.

Adding & Subtracting

Find a common denominator:

$\frac{2}{x+1} + \frac{3}{x-1} = \frac{2(x-1) + 3(x+1)}{(x+1)(x-1)} = \frac{5x + 1}{x^2 - 1}$

Undefined Values (Domain Restrictions)

A rational expression is undefined when the denominator equals zero. The SAT asks: "What value of $x$ makes the expression undefined?"

Worked Example 1 — Multi-step Simplification

Simplify $\frac{2x^2 - 8}{x^2 - x - 2}$.

Worked Example 2 — Subtracting with LCD

$\frac{x}{x+3} - \frac{2}{x-1}$

Rational Expressions 🎯

Complex Fractions

A complex fraction has a fraction in the numerator, denominator, or both:

$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$

Strategy: Multiply the top and bottom by the LCD of all the little fractions.

Worked Example 3 — Simplifying a Complex Fraction

Simplify $\frac{\frac{1}{x} - \frac{1}{3}}{\frac{1}{x} + \frac{1}{3}}$.

Common SAT Trap — Canceling Terms vs. Factors

Rule: You can only cancel common factors — never individual terms.

Advanced Rational Expressions 🎯

Simplification Check 🔍

Is each simplification valid?

Key Takeaways — Part 5

• Matching numerators: when two rational expressions are equal, set the numerators equal and plug in convenient $x$-values to find unknown constants

Polynomials & Factoring

Part 6 of 7 — Polynomial Graphs and Transformations

Reading Polynomial Graphs

From a graph, you can determine:

• Zeros: where the curve crosses/touches the x-axis
• y-intercept: where the curve crosses the y-axis (the constant term)
• Degree: count the number of turns + 1 (approximately)
• Leading coefficient sign: from end behavior

Transformations

For $f(x) = x^3$:

SAT Graph Reading Strategy

When the SAT shows a polynomial graph and asks for the equation:

1. Read the x-intercepts → write factors
2. Check end behavior → determine sign of leading coefficient
3. Check one more point (often the y-intercept) → determine the leading coefficient

Worked Example 1 — From Graph to Equation

A graph crosses at $x = -1$ and $x = 4$, bounces at $x = 2$, and passes through $(0, -16)$. Find the equation.

Worked Example 2 — Transformation Chain

If $f(x) = x^3$, describe the graph of $g(x) = -2(x + 1)^3 + 5$.

The inflection point moves from $(0, 0)$ to $(-1, 5)$.

Polynomial Graphs 🎯

Matching Equations to Graphs — Decision Framework

On the SAT, you'll often see four equation choices and one graph (or vice versa). Here's how to eliminate quickly:

Worked Example 3 — Elimination by y-intercept

Which polynomial has y-intercept $-6$ and zeros at $x = 1, 2, 3$?

Answer: $(x-1)(x-2)(x-3)$ — no extra coefficient needed.

Inside vs. Outside — Transformation Direction

A common SAT trap: shifts inside the function go the opposite direction.

Memory trick: Inside is "opposite" — outside is "obvious."

Graphs & Transformations 🎯

Transformation Identifier 🔍

What transformation does each change represent?

Key Takeaways — Part 6

• On the SAT, always check the y-intercept — it's the fastest way to narrow four answer choices down to one

Polynomials & Factoring

Part 7 of 7 — Review & Advanced SAT Problems

Factoring Decision Tree

1. GCF? Always check first
2. Two terms? → Difference of squares ($a^2 - b^2$) or sum/difference of cubes
3. Three terms? → Trinomial factoring or completing the square
4. Four terms? → Factor by grouping

Factor by Grouping

$x^3 + 3x^2 + 2x + 6$:

• Group: $(x^3 + 3x^2) + (2x + 6)$
• Factor each group: $x^2(x + 3) + 2(x + 3)$
• Factor the common binomial: $(x^2 + 2)(x + 3)$

Special SAT Pattern: Disguised Quadratics

$x^4 - 5x^2 + 4$: let $u = x^2$:

$u^2 - 5u + 4 = (u - 1)(u - 4) = (x^2 - 1)(x^2 - 4) = (x+1)(x-1)(x+2)(x-2)$

This technique works whenever you see $ax^{2n} + bx^n + c$.

Worked Example 1 — Multi-layer Factoring

Factor completely: $3x^3 - 12x$.

Always start with GCF — it reveals hidden patterns.

Worked Example 2 — Algebraic Identity on the SAT

If $a - b = 7$ and $a^2 - b^2 = 35$, find $a + b$.

Advanced Factoring 🎯

Putting It All Together — SAT Strategy

On the SAT, factoring isn't always labeled "factor this." It often appears disguised:

Worked Example 3 — SAT-Style Identity Problem

If $9x^2 - 6x + 1 = 0$, what is the value of $3x - 1$?

Answer: $3x - 1 = 0$. No need to find $x$ at all!

Worked Example 4 — Disguised Difference of Squares

Compute $1003^2 - 997^2$ without a calculator.

SAT Factoring Patterns Cheat Sheet

SAT-Level Challenge 🎯

Name That Factoring Pattern 🔍

Identify which factoring technique applies to each expression.

Key Takeaways — Part 7

Full Topic Summary — Polynomials & Factoring