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Part 1: Polynomial Basics
Polynomials & Factoring
Part 1 of 7 โ Polynomial Basics
What is a Polynomial?
A polynomial is an expression with one or more terms: anโxn+anโ1โxnโ1+โฏ+a1โx+a0โ
- Degree: highest power of x (e.g., 3x4+2xโ1 has degree 4)
- Leading coefficient: coefficient of the highest-degree term
- Constant term: the term with no variable (a)
Adding & Subtracting Polynomials
Combine like terms (same variable and exponent):
(3x2+5xโ2)+(x2
(3x2+5xโ2)โ(x
Subtraction trap: distribute the negative sign to ALL terms in the second polynomial!
Multiplying Polynomials
Use distribution (FOIL for binomials):
(2x+3)(xโ4)=2x2โ8x
Polynomial Operations ๐ฏ
Key Takeaways โ 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 โ Factoring Techniques
GCF Factoring
Always look for a Greatest Common Factor first:
6x3+9x2=3x
Part 3: Special Products
Polynomials & Factoring
Part 3 of 7 โ Polynomial Division
Long Division
Divide x3+2x2โ5x+6 by :
Part 4: Polynomial Division
Polynomials & Factoring
Part 4 of 7 โ 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, the zeros are .
Part 5: Zeros & Roots
Polynomials & Factoring
Part 5 of 7 โ Rational Expressions
Simplifying Rational Expressions
A rational expression is a fraction with polynomials:
x
Part 6: Problem-Solving Workshop
Polynomials & Factoring
Part 6 of 7 โ 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)=x3:
| Transformation | Equation | Effect |
|---|
| Vertical shift up |
Part 7: Review & Applications
Polynomials & Factoring
Part 7 of 7 โ Review & Advanced SAT Problems
Factoring Decision Tree
- GCF? Always check first
- Two terms? โ Difference of squares (a2โb2) or sum/difference of cubes
- Three terms? โ Trinomial factoring or completing the square
- Four terms? โ Factor by grouping
Factor by Grouping
:
0โ
โ
3x+
7)=
4x2+
2x+
5
2
โ
3x+
7)=
3x2+
5xโ
2โ
x2+
3xโ
7=
2x2+
8xโ
9
+
3xโ
12=
2x2โ
5xโ
12
2
(
2
x
+
3)
Difference of Squares
a2โb2=(a+b)(aโb)
Example: x2โ49=(x+7)(xโ7)
Tricky example: 4x2โ25=(2x)2โ52=(2x+5)(2xโ5)
Perfect Square Trinomials
a2+2ab+b2=(a+b)2
a2โ2ab+b2=(aโb)
How to recognize: first and last terms are perfect squares, middle term is ยฑ2รfirstโรlastโ.
x2+10x+25=(x+5)2 because 2(x)(5)=10x โ
Sum/Difference of Cubes (Rare on SAT)
a3+b3=(a+b)(a2โab+b2)
a3โb3=(aโb)(a
Key Takeaways โ Part 2
- Always check for GCF first
- a2โb2=(a+b)(aโb) โ memorize this cold
- Perfect square trinomials: check if middle term = 2ab
- "Factor completely" means keep going until no factor can be factored further
xโ1
- x3รทx=x2. Multiply: x2(xโ1)=x3โx2. Subtract: 3x2โ5x
- 3x2รทx=3x. Multiply: 3x(xโ1)=. Subtract:
- โ2xรทx=โ2. Multiply: โ2(xโ1)=โ2x+2. Subtract:
Result: x2+3xโ2 remainder 4.
Synthetic Division (Faster!)
For dividing by (xโc): write the coefficients, bring down, multiply, add.
Dividing x3+2x2โ5x+6 by (xโ1):
Result: x2+3xโ2 with remainder 4.
The Remainder Theorem
The remainder when f(x) is divided by (xโc) equals f(c).
Check: f(1)=1+2โ5+6=4 โ
Key Takeaways โ Part 3
- Remainder Theorem: remainder of f(x)รท(xโc)=f(c) โ plug in and evaluate!
- Factor Theorem: (xโc) is a factor iff f(c)=0
- Synthetic division is faster than long division for linear divisors (xโc)
- Don't forget to include 0 coefficients for missing terms in division
)
=
(xโ
2)(x+
5)(xโ
1)
x=2,โ5,1 Multiplicity
The multiplicity of a zero is how many times its factor appears.
f(x)=(xโ3)2(x+1):
- x=3 has multiplicity 2 (graph touches x-axis and bounces)
- x=โ1 has multiplicity 1 (graph crosses x-axis)
End Behavior
| Degree | Leading Coeff. | Left End | Right End |
|---|
| Even | Positive | โ | โ |
| Even | Negative | โ | โ |
| Odd | Positive | โ | โ |
| Odd | Negative | โ | โ |
SAT Connection
The SAT asks: "How many x-intercepts does the graph of f(x)=x3โ4x have?"
Factor: x(x2โ4)=x(xโ2)(x+2). Three distinct factors โ 3 x-intercepts.
Zeros & End Behavior ๐ฏ
Key Takeaways โ 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(xโr1โ)(xโr2โ)โฆ and find a from another point
2
+
5
x
+
6
x2โ9
โ
=
(x+2)(x+3)(x+3)(xโ3)โ=
x+2xโ3โ(x๎ =
โ3)
Steps: Factor numerator and denominator, then cancel common factors.
Multiplying & Dividing
Multiply: Factor, cancel, then multiply what remains.
x+1x2โ4โโ
xโ2x+1โ=x+1(x+2)(xโ2)โโ
xโ2x+1โ=x+2
Divide: Flip the second fraction and multiply.
Adding & Subtracting
Find a common denominator:
x+12โ+xโ13โ=(x+1)(xโ1)2(xโ1)+3(x+1)โ=x2โ15x+1โ
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 ๐ฏ
Key Takeaways โ 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
| Horizontal shift right h | f(xโh) | Graph moves right |
| Vertical stretch by a | af(x) | Taller/narrower |
| Reflection over x-axis | โf(x) | Flip upside down |
| Reflection over y-axis | f(โx) | Flip left-right |
SAT Graph Reading Strategy
When the SAT shows a polynomial graph and asks for the equation:
- Read the x-intercepts โ write factors
- Check end behavior โ determine sign of leading coefficient
- Check one more point (often the y-intercept) โ determine the leading coefficient
Key Takeaways โ Part 6
- Read x-intercepts from the graph to write factors
- Use end behavior to determine degree (even/odd) and leading coefficient (ยฑ)
- Transformations: inside the function = horizontal (and reversed); outside = vertical
- (xโh) shifts right, not left โ the shift is opposite the sign
x3+3x2+2x+6
- Group: (x3+3x2)+(2x+6)
- Factor each group: x2(x+3)+2(x+3)
- Factor the common binomial: (x2+2)(x+3)
Special SAT Pattern: Disguised Quadratics
x4โ5x2+4: let u=x2:
u2โ5u+4=(uโ1)(uโ4)=(x2โ1)(x2โ4)=(x+1)(xโ1)(x+2)(xโ2)
This technique works whenever you see ax2n+bxn+c.
Key Takeaways โ Part 7
- Follow the factoring decision tree: GCF โ pattern recognition โ grouping
- Factor by grouping: split into pairs, factor each, extract common binomial
- Disguised quadratics: substitute u=xn to reveal the pattern
- Know the identity (x+y)2=x2+2xy+y2 โ it appears on the SAT frequently
2
2
+
ab+
b2)
3
x2
โ
3x