Notes
Working with polynomials
Write polynomials in standard form (highest degree first). Add by combining like terms; subtract by distributing the negative first. Multiply by distributing every term, and use the special products:
- $(a + b)^2 = a^2 + 2ab + b^2$ and $(a - b)^2 = a^2 - 2ab + b^2$
- $(a + b)(a - b) = a^2 - b^2$
Factoring checklist
1. Take out the GCF. 2. Two terms: is it a difference of squares? 3. Three terms: $x^2 + bx + c$ (find numbers that multiply to $c$ and add to $b$), or $ax^2 + bx + c$ (AC method), or a perfect square trinomial. 4. Four terms: try grouping. 5. Check each factor to see if it factors again, and multiply back to check.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$4x^2 - 3x - x^2 + 5x - 2$.
$x^2 + 2x - 6x - 12$.
$6 \cdot -4 = -24$ and $6 - 4 = 2$.
Take out 3, then factor the difference of squares.
$ac = -30$: 15 and $-2$. $5x^2 + 15x - 2x - 6$.
$x^2 + 8x + 15 = (x + 3)(x + 5)$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A square garden has side length $x$ feet. A designer adds a 2-foot path on every side, making a bigger square.
(a) Write a polynomial in standard form for the area of the bigger square.
Show the answer and scoring guide
Each side is $x + 4$, so the area is $(x + 4)^2 = x^2 + 8x + 16$.
Check off the points you earned:
(b) Write a polynomial for the area of the path alone and simplify.
Show the answer and scoring guide
$(x^2 + 8x + 16) - x^2 = 8x + 16$ square feet.
Check off the points you earned:
(c) Factor your answer from (b) and use it to find the path's area when $x = 10$.
Show the answer and scoring guide
$8x + 16 = 8(x + 2)$. When $x = 10$, $8(12) = 96$ square feet.
Check off the points you earned: