Polynomial Power Guided Notes
Type 5 Focus
Binomial × Trinomial Multiplication
Reference: Example 5 • Target: \((x - 1)(x^2 - 5x - 3)\)
Name: ______________________
Date: ____________ Period: _____
Core Strategy: Double Distributive or Box
Multiply each term in the binomial by every term in the trinomial to generate \(2 \times 3 = 6\) initial terms.
Pro-Tip • Standard Form
Align terms by degree (\(x^3, x^2, x\), constants) and distribute negative signs carefully before combining like terms.
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((x - 1)(x^2 - 5x - 3)\)
\(x(x^2 - 5x - 3) - 1(x^2 - 5x - 3)\)
\(= x^3 - 5x^2 - 3x - x^2 + 5x + 3\)
\(= x^3 + (-5x^2 - x^2) + (-3x + 5x) + 3\)
Final: \(x^3 - 6x^2 + 2x + 3\)
2. Guided Walkthrough We Do
\((x + 2)(x^2 - 4x + 5)\)
Distribute \(x\) and \(+2\):
\(x^3 - 4x^2 + 5x + \underline{\quad\quad\quad}\)
Combine like terms:
\(x^3 + (\underline{\quad})x^2 + (\underline{\quad})x + 10\)
Answer: \(x^3 - 2x^2 - 3x + 10\)
3. Quick Check You Do
\((2x - 3)(x^2 + 3x - 4)\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((x + 4)(x^2 - 3x - 2)\)
Standard Form:
Problem 2: \((x - 5)(2x^2 + x - 3)\)
Standard Form:
Problem 3: \((3x + 1)(x^2 - 2x + 6)\)
Standard Form:
Problem 4: \((2a - 4)(3a^2 - 5a + 1)\)
Standard Form:
Polynomial Power Series • Section 5: Binomial × Trinomial Page 1 of 6
Type 6 Focus
Binomials with Leading Coefficient \(a > 1\)
Reference: Example 6 • Target: \((2x + 1)(x - 7)\)
Name: ______________________
Date: ____________ Period: _____
Core Strategy: FOIL / Double Distribution
Multiply \(\text{First} + \text{Outer} + \text{Inner} + \text{Last}\). Coefficients multiply: \((ax)(cx) = acx^2\).
Middle Term Check
The Outer \((ad)\) and Inner \((bc)\) terms combine into \((ad + bc)x\). Watch sign changes closely!
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((2x + 1)(x - 7)\)
F: \((2x)(x) = 2x^2\)
O: \((2x)(-7) = -14x\)
I: \((1)(x) = +1x\)
L: \((1)(-7) = -7\)
Final: \(2x^2 - 13x - 7\)
2. Guided Walkthrough We Do
\((3x + 2)(2x - 5)\)
F: \((3x)(2x) = 6x^2\)
O + I: \(-15x + 4x = \underline{\quad\quad}\)
L: \((2)(-5) = \underline{\quad\quad}\)
Combine all 4 terms:
Answer: \(6x^2 - 11x - 10\)
3. Quick Check You Do
\((4x - 3)(x + 6)\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((3x + 4)(x - 2)\)
Standard Form:
Problem 2: \((2x - 5)(3x + 1)\)
Standard Form:
Problem 3: \((5y + 2)(2y - 3)\)
Standard Form:
Problem 4: \((4m - 1)(3m - 7)\)
Standard Form:
Polynomial Power Series • Section 6: Binomials with \(a > 1\) Page 2 of 6
Type 7 Focus
Monic Binomials with Leading Coefficient \(a = 1\)
Reference: Example 7 • Target: \((a + 10)(a - 2)\)
Name: ______________________
Date: ____________ Period: _____
Sum-and-Product Shortcut
When leading coefficients are \(1\): \((x + p)(x + q) = x^2 + (p + q)x + pq\).
Quick Mental Check
Middle term = sum of constants \((p + q)\). Last term = product of constants \((p \cdot q)\).
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((a + 10)(a - 2)\)
First: \(a \cdot a = a^2\)
Sum: \(10 + (-2) = +8a\)
Product: \((10)(-2) = -20\)
\(= a^2 + 8a - 20\)
Final: \(a^2 + 8a - 20\)
2. Guided Walkthrough We Do
\((x - 8)(x + 3)\)
First: \(x \cdot x = x^2\)
Sum: \(-8 + 3 = \underline{\quad\quad}\)
Product: \((-8)(3) = \underline{\quad\quad}\)
Combine into trinomial:
Answer: \(x^2 - 5x - 24\)
3. Quick Check You Do
\((y - 6)(y - 4)\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((x + 7)(x + 3)\)
Standard Form:
Problem 2: \((m - 9)(m + 4)\)
Standard Form:
Problem 3: \((p + 12)(p - 5)\)
Standard Form:
Problem 4: \((w - 11)(w - 3)\)
Standard Form:
Polynomial Power Series • Section 7: Monic Binomials (\(a = 1\)) Page 3 of 6
Type 8 Focus
Special Products • Square of a Sum \((A + B)^2\)
Reference: Example 8 • Target: \((2x + 4)^2\)
Name: ______________________
Date: ____________ Period: _____
Formula: \((A + B)^2 = A^2 + 2AB + B^2\)
Square the first term, add twice the product of both terms, and square the last term.
Crucial Error Warning
\((A + B)^2 \neq A^2 + B^2\)! You MUST remember the middle cross-term \(2AB\)!
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((2x + 4)^2\)
\(A = 2x,\quad B = 4\)
\(A^2 = (2x)^2 = 4x^2\)
\(2AB = 2(2x)(4) = 16x\)
\(B^2 = (4)^2 = 16\)
Final: \(4x^2 + 16x + 16\)
2. Guided Walkthrough We Do
\((3x + 5)^2\)
\(A = 3x,\quad B = 5\)
\(A^2 = (3x)^2 = 9x^2\)
\(2AB = 2(3x)(5) = \underline{\quad\quad}\)
\(B^2 = (5)^2 = \underline{\quad\quad}\)
Answer: \(9x^2 + 30x + 25\)
3. Quick Check You Do
\((4x + 1)^2\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((x + 6)^2\)
Standard Form:
Problem 2: \((3x + 2)^2\)
Standard Form:
Problem 3: \((5n + 3)^2\)
Standard Form:
Problem 4: \((6x + 7)^2\)
Standard Form:
Polynomial Power Series • Section 8: Square of a Sum Page 4 of 6
Type 9 Focus
Special Products • Square of a Difference \((A - B)^2\)
Reference: Example 9 • Target: \((3y - 7)^2\)
Name: ______________________
Date: ____________ Period: _____
Formula: \((A - B)^2 = A^2 - 2AB + B^2\)
Square the first term, subtract twice the product of both terms, and square the last term.
Sign Rules to Remember
The middle term is subtracted (\(-2AB\)), but the last term is always positive (\(+B^2\)).
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((3y - 7)^2\)
\(A = 3y,\quad B = 7\)
\(A^2 = (3y)^2 = 9y^2\)
\(-2AB = -2(3y)(7) = -42y\)
\(B^2 = (-7)^2 = +49\)
Final: \(9y^2 - 42y + 49\)
2. Guided Walkthrough We Do
\((2x - 5)^2\)
\(A = 2x,\quad B = 5\)
\(A^2 = (2x)^2 = 4x^2\)
\(-2AB = -2(2x)(5) = \underline{\quad\quad}\)
\(B^2 = (5)^2 = \underline{\quad\quad}\)
Answer: \(4x^2 - 20x + 25\)
3. Quick Check You Do
\((5m - 2)^2\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((x - 9)^2\)
Standard Form:
Problem 2: \((4y - 3)^2\)
Standard Form:
Problem 3: \((6k - 5)^2\)
Standard Form:
Problem 4: \((2a - 9)^2\)
Standard Form:
Polynomial Power Series • Section 9: Square of a Difference Page 5 of 6
Type 10 Focus
Special Products • Difference of Squares \((A - B)(A + B)\)
Reference: Example 10 • Target: \((x - 10)(x + 10)\)
Name: ______________________
Date: ____________ Period: _____
Formula: \((A - B)(A + B) = A^2 - B^2\)
When multiplying conjugate binomials (same terms, opposite middle signs), the middle terms cancel!
The Vanishing Middle Term
Outer \((+AB)\) and Inner \((-AB)\) sum to zero. The result is always a binomial!
Guided Practice • 3 Scaffolded Models
1. Teacher Model I Do
\((x - 10)(x + 10)\)
First: \(x \cdot x = x^2\)
Outer + Inner: \(10x - 10x = 0\)
Last: \((-10)(10) = -100\)
\(= x^2 - 100\)
Final: \(x^2 - 100\)
2. Guided Walkthrough We Do
\((3x - 4)(3x + 4)\)
\(A = 3x,\quad B = 4\)
\(A^2 = (3x)^2 = 9x^2\)
\(B^2 = 4^2 = 16\)
Subtract: \(A^2 - B^2\)
Answer: \(9x^2 - 16\)
3. Quick Check You Do
\((5y + 8)(5y - 8)\)
Ans: ________________________
Independent Practice • 4 Problems (Show All Steps)
Express answers in standard form
Problem 1: \((x + 7)(x - 7)\)
Standard Form:
Problem 2: \((2x - 3)(2x + 3)\)
Standard Form:
Problem 3: \((4a + 9)(4a - 9)\)
Standard Form:
Problem 4: \((7m - 6)(7m + 6)\)
Standard Form:
Polynomial Power Series • Section 10: Difference of Squares Page 6 of 6
