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Polynomials and Factoring

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Polynomials and Factoring - Complete Interactive Lesson

Part 1: Polynomial Basics

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

Step	Work
First two factors	$(x+1)(x-3) = x^2 - 2x - 3$

Worked Example 2 — Subtraction Trap

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

Step	Work
Distribute negative	$5x^3 - x^2 + 4 - 3x^3 - 2x^2 + x - 1$

Polynomial Operations 🎯

Special Products to Memorize

Pattern	Expansion
$(a + b)^2$	$a^2 + 2ab + b^2$

Special Products & Degree 🎯

Identify the Property 🔍

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

Key Takeaways — Part 1

Concept	Key Point
Degree	Highest exponent (not first term!)
Subtraction	Distribute negative to ALL terms
$(a+b)^2$	$a^2 + 2ab + b^2$ (don't forget middle term)

Part 2: Factoring Techniques

Polynomials & Factoring

Part 2 of 7 — Factoring Techniques

GCF Factoring

Always look for a Greatest Common Factor first:

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

Part 3: Special Products

Polynomials & Factoring

Part 3 of 7 — Polynomial Division

Long Division

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

Part 4: Polynomial Division

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 .

Part 5: Zeros & Roots

Polynomials & Factoring

Part 5 of 7 — Rational Expressions

Simplifying Rational Expressions

A rational expression is a fraction with polynomials:

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

Part 6: Problem-Solving Workshop

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 $f(x) = x^3$ :

Transformation	Equation	Effect
Vertical shift up

Part 7: Review & Applications

Polynomials & Factoring

Part 7 of 7 — Review & Advanced SAT Problems

Factoring Decision Tree

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

Factor by Grouping

$x^{3} + 3 x^{2} + 2$ :

Worked Example 3 — Using Special Products

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

Step	Work
Apply $(a - b)^2$	$(3x)^2 - 2(3x)(2) + (2)^2$
Simplify	$9x^2 - 12x + 4$

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

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

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

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

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: $2x^3 - 18x$ .

Step	Work
GCF	$2x(x^2 - 9)$
Difference of squares	$2x(x + 3)(x - 3)$

Worked Example 2 — Recognizing Perfect Squares

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

Check	Result
First term: $(3x)^2$ ?	Yes ✓
Last term: $5^2$ ?	Yes ✓
Middle: $2(3x)(5) = 30x$ ?	Yes ✓
Factor	$(3x - 5)^2$

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 .

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

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

Worked Example 3 — AC Method

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

Step	Work
$a \cdot c$	$6 × (-10) = -60$
Find pair: product $-60$ , sum

Worked Example 4 — Signs Guide

If $c > 0$ , $b > 0$	Both factors positive	$(x + ?)(x + ?)$
If ,

Trinomial Factoring 🎯

What Factoring Method? 🔍

Choose the best factoring technique for each expression.

Key Takeaways — Part 2

Method	When to Use	Example
GCF	Always check first	$6x^3 + 9x^2 = 3x^2(2x+3)$
Diff. of squares	$a^2 - b^2$	$x^2 - 49 = (x+7)(x-7)$
Perfect square	$a^2 \pm 2ab + b^2$	$x^2 + 10x + 25 = (x+5)^2$
Trinomial ( $a=1$ )	Find pair: product $c$ , sum $b$	$x^2 + 7x + 12 = (x+3)(x+4)$
AC method ( $a≠1$ )	Product $ac$ , sum $b$ , then group	$6x^2 + 11x - 10$

"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

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

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 $(x - 1)$ :

	1	2	-5	6
1↓		1	3	-2
	1	3	-2	4

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 $(x - 2)$ .

Step	Work
Include 0 placeholders	Coefficients: $1, 0, 0, -8$
Synthetic with $c = 2$	Bring down 1 → $1 \cdot 2 = 2$ → $0 + 2 = 2$ → $2 \cdot 2 = 4$ → $0 + 4 = 4$ → $4 \cdot 2 = 8$ → $-8 + 8 = 0$
Result	$x^2 + 2x + 4$ , remainder $0$

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 $2x^3 + x^2 - 7x + 2$ ?

Step	Work
$(x + 2) = (x - (-2))$ , so $c = -2$
Evaluate $f(-2)$	$2(-8) + (4) - 7(-2) + 2$
Simplify	$-16 + 4 + 14 + 2 = 4$
Remainder	$4 \neq 0$ , so NO — not a factor

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.

Step	Work
Divide	$f(1) = 1 - 4 + 7 = 4$ → remainder is $4$
Synthetic:

SAT Shortcut: Remainder without Division

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

Division Applications 🎯

Remainder Theorem Quick Check 🔍

Find the remainder without doing long division.

Key Takeaways — Part 3

Tool	Purpose	Speed
Long division	Any divisor	Slow
Synthetic division	Divisor $(x - c)$ only	Fast
Remainder Theorem	Find remainder only	Fastest
Factor Theorem	Check if factor	Fastest

Remainder Theorem: $f(x) ÷ (x-c)$ → 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: $f(x) = d(x) \cdot q(x) + r$

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)

Degree	Leading Coeff.	Left End	Right End
Even	Positive	↑	↑
Even	Negative	↓	↓
Odd	Positive	↓	↑
Odd	Negative	↑	↓

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 $3$ .

Step	Work
Write factors	$(x + 1)(x - 2)(x - 5)$
Apply leading coeff.	$3(x + 1)(x - 2)(x - 5)$

Worked Example 2 — Finding $a$ from a Point

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

Step	Work
Substitute $(2, 10)$	$10 = a(2 - 1)(2 + 3)$
Simplify	$10 = a(1)(5) = 5a$
Solve	$a = 2$
Answer	$f(x) = 2(x - 1)(x + 3)$

Zeros & End Behavior 🎯

Multiplicity and Graph Behavior

Multiplicity	Graph at that zero	Example
1 (odd)	Crosses x-axis	$f(x) = x - 2$ at $x = 2$
2 (even)	Touches and bounces	$f(x) = (x - 2)^2$ at $x = 2$
3 (odd)	Crosses with inflection	$f(x) = (x - 2)^3$ at $x = 2$

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?

Zero	Min. multiplicity
$x = -3$ (crosses)	1
$x = 1$ (bounces)	2
$x = 4$ (crosses)	1

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?

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

Graph Analysis 🎯

Match the Graph Feature 🔍

What does each piece of information tell you?

Key Takeaways — Part 4

Concept	Key Rule
Zeros from factors	$(x - r)$ → zero at $x = r$
Even multiplicity	Graph bounces at zero
Odd multiplicity	Graph crosses at zero
End behavior	Degree (even/odd) + sign of leading coeff.
Max turning points	Degree minus 1
Build polynomial	$f(x) = a(x - r_1)(x - r_2)\cdots$

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

(x+2)(x+3)(x+3)(x−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 $\frac{2x^2 - 8}{x^2 - x - 2}$ .

Step	Work
Factor numerator	$2(x^2 - 4) = 2(x+2)(x-2)$
Factor denominator	$(x-2)(x+1)$
Cancel $(x-2)$	$\frac{2(x+2)}{x+1}$ ,

Worked Example 2 — Subtracting with LCD

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

Step	Work
LCD	$(x+3)(x-1)$
Rewrite	$\frac{x(x-1) - 2(x+3)}{(x+3)(x-1)}$
Expand	$\frac{x^2 - x - 2x - 6}{(x+3)(x-1)} = \frac{x^2 - 3x - 6}{(x+3)(x-1)}$

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

Step	Work
LCD of inner fractions	$3x$
Multiply top and bottom by $3x$	$\frac{3x \cdot \frac{1}{x} - 3x \cdot \frac{1}{3}}{3x \cdot \frac{1}{x} + 3x \cdot \frac{1}{3}}$

Common SAT Trap — Canceling Terms vs. Factors

Expression	Can you cancel?	Why?
$\frac{(x+2)(x-3)}{(x+2)}$	✅ Yes	is a of both

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

Advanced Rational Expressions 🎯

Simplification Check 🔍

Is each simplification valid?

Key Takeaways — Part 5

Operation	Procedure
Simplify	Factor top & bottom → cancel common factors
Multiply	Factor all → cancel across → multiply
Divide	Flip 2nd fraction → multiply
Add/Subtract	Find LCD → combine numerators
Complex fraction	Multiply top & bottom by LCD of inner fractions

Common Trap	Fix
Canceling terms ( $\frac{x+2}{x+5}$ )	Only cancel factors
Forgetting restrictions	State $x \neq$ (zeros of original denominator)
Sign errors in subtraction	Distribute the minus to ALL terms

Partial fractions (splitting one fraction into two) appear on harder SAT problems — reverse the adding process

SAT Graph Reading Strategy

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

Read the x-intercepts → write factors
Check end behavior → determine sign of leading coefficient
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.

Step	Work
Write factors	$a(x + 1)(x - 4)(x - 2)^2$
Use $(0, -16)$	$a(1)(-4)(4) = -16a = -16$
Solve	$a = 1$
Answer	$f(x) = (x+1)(x-4)(x-2)^2$

Worked Example 2 — Transformation Chain

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

Transformation	Rule	Effect
$(x + 1)^3$	$f(x - h)$ with $h = -1$	Shift left 1
$2(\cdots)$	$af(x)$ with $a = 2$	Vertical stretch by 2
$-(\cdots)$	$-f(x)$	Reflect over x-axis
$+ 5$	$f(x) + k$	Shift up 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:

Check	What it tells you	How to read it
End behavior	Degree (even/odd) + sign	Both ends same = even; opposite = odd
x-intercepts	Factors and their multiplicity	Crosses = odd mult.; bounces = even
y-intercept	Constant term	Plug $x = 0$ into each answer choice
Number of turns	Approximate degree	Turns ≤ degree − 1

Worked Example 3 — Elimination by y-intercept

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

Option	y-int (plug $x = 0$ )	Match?
$(x-1)(x-2)(x-3)$

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.

Written	Direction
$f(x - 3)$	Right 3
$f(x + 3)$	Left 3
$f(x) - 3$

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

Graphs & Transformations 🎯

Transformation Identifier 🔍

What transformation does each change represent?

Key Takeaways — Part 6

Skill	Strategy
Graph → Equation	Read zeros, end behavior, y-intercept
Equation → Graph	Plot zeros, check multiplicity, draw end behavior
Transformations	Inside = horizontal (opposite); outside = vertical
Elimination	Plug $x = 0$ into choices to match y-intercept

Transformation	Direction Rule
$f(x - h)$	Right $h$ (opposite sign)
$f(x + h)$	Left $h$ (opposite sign)
$f(x) + k$	Up $k$ (same sign)
$af(x)$ , $a > 1$	Vertical stretch
$-f(x)$	Reflect over x-axis

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

x^{3} + 3 x^{2} + 2 x + 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: $3x^3 - 12x$ .

Step	Work
GCF first	$3x(x^2 - 4)$
Diff. of squares	$3x(x + 2)(x - 2)$

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 $a + b$ .

Step	Work
Recognize identity	$a^2 - b^2 = (a+b)(a-b)$
Substitute	$35 = (a+b)(7)$
Solve	$a + b = 5$

Advanced Factoring 🎯

Putting It All Together — SAT Strategy

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

SAT Question Type	Factoring Skill Needed
"Simplify the expression"	Factor and cancel
"How many solutions?"	Factor, count zeros
"What is the value of...?"	Factor to reveal identity
"Which is equivalent?"	Factor and match
"Find the zeros"	Factor and solve

Worked Example 3 — SAT-Style Identity Problem

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

Step	Work
Recognize	$9x^2 - 6x + 1 = (3x - 1)^2$

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.

Step	Work
Identity	$a^2 - b^2 = (a+b)(a-b)$

SAT Factoring Patterns Cheat Sheet

Pattern	Formula	Example
Difference of squares	$a^2 - b^2 = (a+b)(a-b)$

SAT-Level Challenge 🎯

Name That Factoring Pattern 🔍

Identify which factoring technique applies to each expression.

Key Takeaways — Part 7

Technique	When to Use	Key Move
GCF	Always first	Factor out common factor
Diff. of squares	$a^2 - b^2$	$(a+b)(a-b)$
Perfect square	$a^2 \pm 2ab + b^2$	$(a \pm b)^2$
Trinomial	$ax^2 + bx + c$	Find factors of $ac$ that add to $b$
Grouping	4 terms	Pair, factor, extract binomial
Disguised quad.	Even powers like $x^4, x^6$	Let $u = x^n$
Identities	"Find $a+b$ " or "Find $x^2 + 1/x^2$ "	Expand or factor known identity

Full Topic Summary — Polynomials & Factoring

Part	Core Skill
1	Polynomial basics, adding/multiplying, special products
2	Factoring techniques: GCF, diff. of squares, trinomials
3	Polynomial division: long division, synthetic, remainder theorem
4	Zeros, multiplicity, end behavior, building from roots
5	Rational expressions: simplify, add, multiply, restrictions
6	Graphs, transformations, matching equation to graph
7	Review: decision tree, identities, SAT strategy

(x+3)(x−1)x2−3x−6

Polynomials and Factoring

Polynomials and Factoring - Complete Interactive Lesson

Part 1: Polynomial Basics

Polynomials & Factoring

What is a Polynomial?

Adding & Subtracting Polynomials

Multiplying Polynomials

Worked Example 1 — Multiplying Three Factors

Worked Example 2 — Subtraction Trap

Special Products to Memorize

Key Takeaways — Part 1

Part 2: Factoring Techniques

Polynomials & Factoring

GCF Factoring

Part 3: Special Products

Polynomials & Factoring

Long Division

Part 4: Polynomial Division

Polynomials & Factoring

Zeros = Roots = x-intercepts

Part 5: Zeros & Roots

Polynomials & Factoring

Simplifying Rational Expressions

Part 6: Problem-Solving Workshop

Polynomials & Factoring

Reading Polynomial Graphs

Transformations

Part 7: Review & Applications

Polynomials & Factoring

Factoring Decision Tree

Factor by Grouping

Worked Example 3 — Using Special Products

Degree of a Product

Difference of Squares

Perfect Square Trinomials

Sum/Difference of Cubes (Rare on SAT)

Worked Example 1 — Multi-Step Factoring

Worked Example 2 — Recognizing Perfect Squares

Factoring Trinomials: ax2+bx+cax^2 + bx + cax2+bx+c

Worked Example 3 — AC Method

Worked Example 4 — Signs Guide

Key Takeaways — Part 2

Synthetic Division (Faster!)

The Remainder Theorem

Worked Example 1 — Missing Term Trap

Worked Example 2 — Using Remainder Theorem Strategically

Writing the Result of Division

Worked Example 3

SAT Shortcut: Remainder without Division

Key Takeaways — Part 3

Multiplicity

End Behavior

SAT Connection

Worked Example 1 — Building a Polynomial from Zeros

Worked Example 2 — Finding aaa from a Point

Multiplicity and Graph Behavior

Worked Example 3 — Degree from Graph

Worked Example 4 — Number of Real Zeros

Key Takeaways — Part 4

Multiplying & Dividing

Adding & Subtracting

Undefined Values (Domain Restrictions)

Worked Example 1 — Multi-step Simplification

Worked Example 2 — Subtracting with LCD

Complex Fractions

Worked Example 3 — Simplifying a Complex Fraction

Common SAT Trap — Canceling Terms vs. Factors

Key Takeaways — Part 5

SAT Graph Reading Strategy

Worked Example 1 — From Graph to Equation

Worked Example 2 — Transformation Chain

Matching Equations to Graphs — Decision Framework

Worked Example 3 — Elimination by y-intercept

Inside vs. Outside — Transformation Direction

Key Takeaways — Part 6

Special SAT Pattern: Disguised Quadratics

Worked Example 1 — Multi-layer Factoring

Worked Example 2 — Algebraic Identity on the SAT

Putting It All Together — SAT Strategy

Worked Example 3 — SAT-Style Identity Problem

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

Worked Example 2 — Finding $a$ from a Point