Station 1 – Recognizing patterns in multiplying binomials and factoring trinomials
We have learned two special patterns for multiplying binomials. FOIL is not one of these
patterns.
1. What are the two special patterns for multiplying binomials? Name them and give
examples of each
These special patterns help us with their counterpart factoring patterns.
2. Name the two factoring patterns that partner with the multiplying patterns and give
examples of each.
Station 1 key
1. Squaring a binomial
Sum and Difference
(𝟐𝒙 − 𝟓)𝟐 = 𝟒𝒙𝟐 − 𝟐𝟎𝒙 + 𝟐𝟓
(𝟒𝒙 + 𝟔)(𝟒𝒙 − 𝟔) = 𝟏𝟔𝒙𝟐 − 𝟑𝟔
2. Perfect square trinomial
Difference of two squares
𝟐𝟓𝒙𝟐 − 𝟑𝟎𝒙 + 𝟗 = (𝟓𝒙 − 𝟑)𝟐
𝟒𝟗𝒙𝟐 − 𝟏𝟐𝟏 = (𝟕𝒙 − 𝟏𝟏)(𝟕𝒙 + 𝟏𝟏)
Station 2 – Simplifying and naming polynomials
Simplify and write in standard form, then name each based on degree and number of terms.
1. (𝟏𝟔𝒄𝟐 + 𝟒𝒄) − 𝟐(𝟑𝒄 + 𝟒𝒄𝟐 ) + (𝒄 + 𝟐)(𝒄 − 𝟒)
2. (𝟏𝟔𝒆𝟑 + 𝟏𝟐𝒆) − (𝟒𝒆𝟒 − 𝟐𝒆𝟑 ) + 𝒆(𝟐 + 𝒆)
3. −𝟑𝒘𝟐 (𝒘 + 𝟏𝟏) − (𝟐𝒘 + 𝟑)(𝟑𝒘 − 𝟓)
4. 𝟏𝟓𝒅 + 𝟏𝟑𝒅𝟐 − 𝟏𝟕𝒅 + 𝟏𝟎𝒅𝟐 − (𝟐𝒅 + 𝟒𝒅𝟐 )
Station 2 key
1. 𝟏𝟔𝒄𝟐 + 𝟒𝒄 − 𝟔𝒄 − 𝟖𝒄𝟐 + 𝒄𝟐 − 𝟒𝒄 + 𝟐𝒄 − 𝟖 = 𝟗𝒄𝟐 − 𝟒𝒄 − 𝟖 ; 2nd degree trinomial
2. 𝟏𝟔𝒆𝟑 + 𝟏𝟐𝒆 − 𝟒𝒆𝟒 + 𝟐𝒆𝟑 + 𝟐𝒆 + 𝒆𝟐 = −𝟒𝒆𝟒 + 𝟏𝟖𝒆𝟑 + 𝒆𝟐 + 𝟏𝟒𝒆 ; 4th degree polynomial
3. −𝟑𝒘𝟑 − 𝟑𝟑𝒘𝟐 − (𝟔𝒘𝟐 − 𝟏𝟎𝒘 + 𝟗𝒘 − 𝟏𝟓) = −𝟑𝒘𝟑 − 𝟑𝟗𝒘𝟐 + 𝒘 + 𝟏𝟓 ; 3rd degree poly.
4. 𝟏𝟓𝒅 + 𝟏𝟑𝒅𝟐 − 𝟏𝟕𝒅 + 𝟏𝟎𝒅𝟐 − (𝟐𝒅 + 𝟒𝒅𝟐 ) = 𝟏𝟗𝒅𝟐 − 𝟒𝒅 ; 2nd degree binomial
Station 3 – GCF Factoring
Factor the following using the GCF. Use standard form.
1. 𝟏𝟎𝒄𝟑 + 𝟒𝒄𝟐 − 𝟔𝒄𝟒
2. 𝟐𝟒𝒂𝟑 𝒃𝟐 + 𝟏𝟔𝒂𝟒 𝒃𝟑 + 𝟖𝒂𝟐 𝒃𝟓
3. −𝟐𝟎𝒙𝟒 + 𝟏𝟓𝒙𝟔 − 𝟒𝟎𝒙𝟑
4. 𝟏𝟐𝒚𝟐 − 𝟐𝟔𝒙𝒚𝟑 + 𝟏𝟎𝒙𝟐 𝒚𝟒
Station 3 key
1. 𝟐𝒄𝟐 (−𝟑𝒄𝟐 + 𝟓𝒄 + 𝟐) = −𝟐𝒄𝟐 (𝟑𝒄𝟐 − 𝟓𝒄 − 𝟐)
2. 𝟖𝒂𝟐 𝒃𝟐 (𝒃𝟑 + 𝟐𝒂𝟐 𝒃 + 𝟑𝒂)
3. −𝟓𝒙𝟑 (−𝟑𝒙𝟑 + 𝟒𝒙 + 𝟖) = 𝟓𝒙𝟑 (𝟑𝒙𝟑 − 𝟒𝒙 − 𝟖)
4. 𝟐𝒚𝟐 (𝟔 − 𝟏𝟑𝒙𝒚 + 𝟓𝒙𝟐 𝒚𝟐 ) = 𝟐𝒚𝟐 (𝟓𝒙𝟐 𝒚𝟐 − 𝟏𝟑𝒙𝒚 + 𝟔)
Station 4 - Multiplying polynomials
Find the product:
1. (𝟕𝒂 + 𝟐𝒃)(−𝟒𝒂 − 𝟑𝒃)
4. (𝟓𝒗 + 𝟐)(𝟓𝒗 − 𝟐)
2. (𝟏𝟐𝒚 − 𝟑)(𝟓𝒘 + 𝟒)
5. (𝟏𝟎𝒑 − 𝟒)𝟐
3. (𝟑𝒘 − 𝟒)(𝟓𝒘𝟐 − 𝟐𝒘 + 𝟏𝟎)
6. (𝟗𝒚 − 𝟓)(𝟐𝒚 − 𝟕)
Station 4 key
1.
−𝟐𝟖𝒂𝟐 − 𝟐𝟗𝒂𝒃 − 𝟔𝒃𝟐
4.
𝟐𝟓𝒗𝟐 − 𝟒
2.
𝟔𝟎𝒘𝒚 + 𝟒𝟖𝒚 − 𝟏𝟓𝒘 − 𝟏𝟐
5.
𝟏𝟎𝟎𝒑𝟐 − 𝟖𝟎𝒑 + 𝟏𝟔
3.
𝟏𝟓𝒘𝟑 − 𝟐𝟔𝒘𝟐 + 𝟑𝟖𝒘 − 𝟒𝟎
6.
𝟏𝟖𝒚𝟐 − 𝟓𝟑𝒚 − 𝟑𝟓
Station 5 – Applications of perimeter and area
For the rectangle below, write the simplified expressions that represent the perimeter and area
of the rectangle.
4𝑥 − 1
2𝑥 + 3
Station 5 key
Perimeter:
𝟐(𝟐𝒙 + 𝟑) + 𝟐(𝟒𝒙 − 𝟏) = 𝟒𝒙 + 𝟔 + 𝟖𝒙 − 𝟐 = 𝟏𝟐𝒙 + 𝟒
Area:
(𝟐𝒙 + 𝟑)(𝟒𝒙 − 𝟏) = 𝟖𝒙𝟐 − 𝟐𝒙 + 𝟏𝟐𝒙 − 𝟑 = 𝟖𝒙𝟐 + 𝟏𝟎𝒙 − 𝟑
Station 6 – Factoring
Factor the following trinomials:
1. 𝒘𝟐 − 𝟏𝟓𝒘 + 𝟒𝟒
5. 𝟏𝟐𝒙𝟐 − 𝟕𝒙 − 𝟏𝟎
2. 𝒑𝟐 + 𝟐𝟑𝒑 + 𝟐𝟐
6. 𝟐𝒙𝟐 − 𝟏𝟗𝒙 + 𝟑𝟗
3. 𝒓𝟐 + 𝟏𝟓𝒓 + 𝟓𝟎
7. 𝟑𝒙𝟐 − 𝟑𝒙 − 𝟔
4. 𝒒𝟐 − 𝟏𝟓𝒒 − 𝟑𝟒
8. 𝟓𝒙𝟐 − 𝟔𝒙 + 𝟏
Station 6 key
1. (𝒘 − 𝟏𝟏)(𝒘 − 𝟒)
5. (𝟒𝒙 − 𝟓)(𝟑𝒙 + 𝟐)
2. (𝒑 + 𝟐𝟐)(𝒑 + 𝟏)
6. (𝟐𝒙 − 𝟏𝟑)(𝒙 − 𝟑)
3. (𝒓 + 𝟏𝟎)(𝒓 + 𝟓)
7. (𝟑𝒙 + 𝟑)(𝒙 − 𝟐)
4. (𝒒 − 𝟏𝟕)(𝒒 + 𝟐)
8. (𝟓𝒙 − 𝟏)(𝒙 − 𝟏)
Station 7 - Dividing Polynomials
Using long division, divide 2x3 – 9x2 + 15 by 2x – 5.
Show all work and express any remainder as a fraction.
Station 7 key
Remember that since there is no “x term” you must fill in with Ox
so the answer is
Station 8 – Finding area using polynomial rules
Write a simplified expression to represent the shaded area below:
𝟑𝒙 + 𝟓
𝟒𝒙 − 𝟐
𝟐𝒙 + 𝟏
Station 8 key
Area of rectangle
(𝟑𝒙 + 𝟓)(𝟒𝒙 − 𝟐) = 𝟏𝟐𝒙𝟐 − 𝟔𝒙 + 𝟐𝟎𝒙 − 𝟏𝟎 = 𝟏𝟐𝒙𝟐 + 𝟏𝟒𝒙 − 𝟏𝟎
Area of triangle
(𝟒𝒙 − 𝟐)(𝟐𝒙 + 𝟏) 𝟖𝒙𝟐 − 𝟒𝒙 + 𝟒𝒙 − 𝟐 𝟖𝒙𝟐 − 𝟐
=
=
= 𝟒𝒙𝟐 − 𝟏
𝟐
𝟐
𝟐
Now subtract and simplify
(𝟏𝟐𝒙𝟐 + 𝟏𝟒𝒙 − 𝟏𝟎) − (𝟒𝒙𝟐 − 𝟏) = 𝟖𝒙𝟐 + 𝟏𝟒𝒙 − 𝟗