🎯⭐ INTERACTIVE LESSON

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):

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

Polynomial Operations 🎯

Key Takeaways — Part 1

Degree = highest exponent; leading coefficient = coefficient of that term
When subtracting polynomials, distribute the negative to EVERY term
Multiply polynomials using distribution (FOIL is just distribution for two binomials)
Combine like terms as the final step

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

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

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)

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

Factoring Patterns 🎯

Key Takeaways — Part 2

Always check for GCF first
$a^2 - b^2 = (a+b)(a-b)$ — memorize this cold
Perfect square trinomials: check if middle term = $2ab$
"Factor completely" means keep going until no factor can be factored further

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

Polynomial Division 🎯

Key Takeaways — Part 3

Remainder Theorem: remainder of $f(x) ÷ (x-c) = f(c)$ — plug in and evaluate!
Factor Theorem: $(x - c)$ is a factor iff $f(c) = 0$
Synthetic division is faster than long division for linear divisors $(x - c)$
Don't forget to include $0$ coefficients for missing terms in division

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.

Zeros & End Behavior 🎯

Key Takeaways — Part 4

Zeros = roots = x-intercepts: set $f(x) = 0$ and solve
Multiplicity: even = bounce off axis; odd = cross through
End behavior: determined by degree and sign of leading coefficient
Given zeros, write $f(x) = a(x - r_1)(x - r_2)\ldots$ and find $a$ from another point

(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?"

Rational Expressions 🎯

Key Takeaways — Part 5

Always factor first before simplifying rational expressions
Cancel only common factors (not terms!)
Undefined when denominator = 0
To add/subtract fractions: find common denominator, combine numerators

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

Polynomial Graphs 🎯

Key Takeaways — Part 6

Read x-intercepts from the graph to write factors
Use end behavior to determine degree (even/odd) and leading coefficient (±)
Transformations: inside the function = horizontal (and reversed); outside = vertical
$(x-h)$ shifts right, not left — the shift is opposite the sign

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 $ax^{2n} + bx^n + c$ .

Advanced Factoring 🎯

Key Takeaways — Part 7

Follow the factoring decision tree: GCF → pattern recognition → grouping
Factor by grouping: split into pairs, factor each, extract common binomial
Disguised quadratics: substitute $u = x^n$ to reveal the pattern
Know the identity $(x+y)^2 = x^2 + 2xy + y^2$ — it appears on the SAT frequently

Polynomials and Factoring

Polynomials and Factoring - Complete Interactive Lesson

Part 1: Polynomial Basics

Polynomials & Factoring

What is a Polynomial?

Adding & Subtracting Polynomials

Multiplying Polynomials

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

Difference of Squares

Perfect Square Trinomials

Sum/Difference of Cubes (Rare on SAT)

Key Takeaways — Part 2

Synthetic Division (Faster!)

The Remainder Theorem

Key Takeaways — Part 3

Multiplicity

End Behavior

SAT Connection

Key Takeaways — Part 4

Multiplying & Dividing

Adding & Subtracting

Undefined Values (Domain Restrictions)

Key Takeaways — Part 5

SAT Graph Reading Strategy

Key Takeaways — Part 6

Special SAT Pattern: Disguised Quadratics

Key Takeaways — Part 7