Notes
Operations on polynomials
- Add by combining like terms. When subtracting, distribute the negative to every term of the second polynomial.
- Multiply by distributing every term: $(x + 3)(x - 5) = x^2 - 2x - 15$. Squaring needs three terms: $(x + 4)^2 = x^2 + 8x + 16$.
- The degree of a product is the sum of the degrees.
Factoring
- Always take out the GCF first. For four terms, try grouping.
- For $x^2 + bx + c$ find two numbers that multiply to $c$ and add to $b$. For $ax^2 + bx + c$ use the ac method.
- Check every factoring by multiplying back out.
Special products
- Difference of squares: $a^2 - b^2 = (a - b)(a + b)$.
- Perfect squares: $a^2 \pm 2ab + b^2 = (a \pm b)^2$.
- A sum of squares does not factor over the reals.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Distribute the negative: $4x^2 - 3x + 1 - x^2 - 2x + 5$. Then combine like terms.
$3x^2 - 12x + 2x - 8$, and $-12x + 2x = -10x$.
$-5 \cdot 2 = -10$ and $-5 + 2 = -3$.
This is $(5x)^2 - 6^2$, a difference of squares.
Factor out $3x$, then factor $x^2 - 4$. The first choice is not fully factored.
$6x^2 - 3x + 4x - 2 = 6x^2 + x - 2$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A rectangle has length $2x + 3$ and width $x + 4$ (in feet).
(a) Write the area as a polynomial in standard form.
Show the answer and scoring guide
$(2x + 3)(x + 4) = 2x^2 + 8x + 3x + 12 = 2x^2 + 11x + 12$.
Check off the points you earned:
(b) Write the perimeter as a polynomial.
Show the answer and scoring guide
Length plus width is $3x + 7$, so the perimeter is $2(3x + 7) = 6x + 14$.
Check off the points you earned:
(c) Find the area and perimeter when $x = 5$, and check the area using the factored form.
Show the answer and scoring guide
Area $= 2(25) + 55 + 12 = 117$ square feet. Perimeter $= 30 + 14 = 44$ feet. Check: $(2 \cdot 5 + 3)(5 + 4) = 13 \cdot 9 = 117$.
Check off the points you earned: