Week 29: Grouping, lines through two points, and radical equations
Field Book
105 to 108
Due
Test 26 due
Method
Paper first, every step
Factoring by grouping handles the polynomials that have four terms and no obvious common factor. Writing a line through two points is the last piece of the line toolkit: find the slope first, then use one of the points to find where it crosses. Radical equations come with a warning attached, because squaring both sides can manufacture answers that were never true.
What you have to be able to do
Factor a four-term polynomial by grouping
Write the equation of a line through two given points, or parallel to a given line
Solve a radical equation and check for extraneous answers
The procedures
Factoring by grouping
Split the four terms into two pairs.
Factor the common factor out of each pair separately.
If the leftover parentheses match, factor that whole parenthesis out. If they do not match, try pairing the terms differently.
A line through two points
Find the slope first using rise over run between the two points.
Put that slope and either one of the points into y = mx + b, and solve for b.
Write the finished equation. Parallel lines have the same slope and a different b.
Radical equations
Get the radical by itself on one side.
Square both sides to remove it, then solve normally.
Check every answer in the original equation. Squaring can create answers that do not actually work, and those are thrown out.
Worked
WORKED
Factor x3 + 2x2 + 3x + 6 by grouping. Pair them: (x3 + 2x2) and (3x + 6). Factor each pair: x2(x + 2) and 3(x + 2). The parentheses match. ANSWER: (x2 + 3)(x + 2)
WORKED
Write the line through (1, 3) and (3, 7). Slope: rise 4 over run 2, which is 2. Use (1, 3): 3 = 2(1) + b, so b = 1. ANSWER: y = 2x + 1
Terms
Term
Meaning
grouping
factoring four terms by splitting them into two pairs
extraneous solution
an answer produced by the algebra that fails in the original equation
parallel lines
lines with the same slope and different intercepts
Check yourself
1. Factor x3 + 5x2 + 2x + 10
2. Line through (0, 1) and (2, 5)
3. Solve: the square root of x equals 5
reveal answers
(x2 + 2)(x + 5) · y = 2x + 1 · x = 25
This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.
-- THE UNKNOWN -- week 29, grouping, lines through two points, and radical equationsalgebra 1 · oda