Tuesday, 1 September 2015
Algebraic expression review
Expanding algebraic expressions
Distributive property
◼a(b + c) = a · b + a · c ◼(b + c) a = b · a + c · a
Special expansion properties
◼(A + B) (A - B) = A2- B2 ◼(A + B)2= A2+ 2 A B + B2 ◼(A - B)2= A2- 2 A B + B2 ◼(A + B)3= A3+ 3 A2B+ 3 A B2+ B3 ◼(A - B)3= A3- 3 A2B+ 3 A B2- B3
Questions
Use the above to expand the following algebraic expressions.
1. (x + 1) (x - 2) 2. (x - 4) (x + 4)
3. 2 x + 3 2 x - 3 4. x2+ 82
Questions
Consider the algebraic expression (x - 1) (x - 2) (x + 2). Expand this to answer the following questions.
1. This is a polynomial. What is its degree? 2. What is the coefficient of the x2 term?
Questions
Expand the following to get two more special formulas.
◼(A - B) A2+ A B + B2 ◼(A + B) A2- A B + B2
Why Factor?
Cubic splines
In computer graphics and graphic design, curves are drawn using third degree polynomials, called cubic splines. The various polynomials must piece together so that there are no gaps. In other words each polynomial in the sequence of pieces must intersect with its neighbor polynomial.
Questions
◼ How would you determine for which values of x the lines y = 2 x + 1 and y = -x + 5 intersect? What is
the point of intersection of these two lines?
◼ How would you determine for which values of x the cubic polynomials y = x3- 3 x2+ 4 x + 2 and
y= 2 x3+ 5 x2+ 4 x + 2 intersect?
Profit
A company manufactures and sells iPhone covers. Their manufacturing plant building cost $520,000. The materials for each cover cost $4.20. Also, personnel and energy costs vary depending on the quantities made. The manufacturing engineer approximates that to make x covers, personnel and energy costs will be -0.02 x2+ 6.33 x dollars. The company sells each cover for $18.99.
Questions
◼ What is an algebraic expression that gives the profit for making and selling x covers?
◼ How could one use this expression to determine the break even point, i.e. the number of covers to
Inverse operations
Questions
◼ What number added to 5 is 19?
◼ What is the opposite operation to addition? ◼ What number multiplied by 4 is 48?
◼ What is the opposite operation to multiplication? ◼ What is the number whose square is 81? ◼ What is the opposite operation to squaring?
Factoring
Factoring is the inverse operation to expanding a polynomial. Factoring is a useful way to find solutions to algebraic equations.
Technique 1
Pull out what is common to all terms. (This again is the distributive property.) For example, to factor the expanded expression 4 x2- 6 x, each term has factor 2 x.
4 x2- 6 x = 2 x(2 x - 3)
Questions
◼ How does 3 h3+ 9 h2 factor?
◼ How does 5 a2b3- 7 a4b2 factor?
Technique 2
We have some special expansion formula that we can use in reverse to factor particular expressions.
◼(A + B) (A - B) = A2- B2 ◼(A + B)2= A2+ 2 A B + B2 ◼(A - B)2= A2- 2 A B + B2 ◼(A + B)3= A3+ 3 A2B+ 3 A B2+ B3 ◼(A - B)3= A3- 3 A2B+ 3 A B2- B3 ◼(A - B) A2+ A B + B2 = A3- B3 ◼(A + B) A2- A B + B2 = A3+ B3
Questions
◼ How does 9 x2- 4 factor?
◼ How does x3- 6 x2+ 12 x - 8 factor?
◼ How does y3+ 64 factor?
Technique 3
Use trial and error focusing on the coefficients of the highest degree term and the constant term.
Example
To factor 2 x2- x - 1 recognize that the constant term -1 can be written as 1×(-1). The coefficient of the
highest degree term 2 x2 is 2 and the simplest way to write 2 as a product is 1×2. Try expanding each
of
(2 x - 1) (x + 1) (2 x + 1) (x - 1)
to see if one of them is 2 x2- x - 1. If so, you have the factorization.
Questions
◼ What is the factorization of x2- 8 x + 12? ◼ What is the factorization of 4 x2+ 4 x - 3?
◼ What is the factorization of 5 x3- 15 x2+ 10 x? (Combine above techniques)
Technique 4
Recognize the expression as a simpler expression after a replacement is made.
Example
To factor (b + 2)2+ 6 (b + 2) + 5, not that this is the same as x2+ 6 x + 5 with x = b + 2. Factor x2+ 6 x + 5
as (x + 5) (x + 1) using the third technique and put back b + 2 in place of x to get (b + 2)2+ 6 (b + 2) + 5 = ((b + 2) + 7) ((b + 2) + 1)
= (b + 9) (b + 3)
Questions
◼ What is the factorization of (3 x - 1)2- 4 (3 x - 1) + 3? ◼ What is the factorization of (5 y + 7)2- (2 y - 3)2?
Try factoring by breaking the expressions into groups.
Example
To factor x4- 4 x3+ x - 4, recognize that the first two terms are similar to the last two terms. Factor out
what is common to just the first to terms. x4- 4 x3+ x - 4 = x3(x - 4) + (x + 4) Now factor out the common x - 4 factor.
x4- 4 x3+ x - 4 = x3(x - 4) + (x + 4) = (x + 4) x3+ 1
Recognize the last cubic factor as one of the special forms, a sum of cubes. x4- 4 x3+ x - 4 = x3(x - 4) + (x + 4) = (x + 4) x3+ 1
(x + 4) (x + 1) x2- x + 1
Questions
1. Factor x4y3- x2y5.
2. Factor 2 x3/2+ 4 x1/2+ 2 x-1/2. (Hint: when you factor out common terms, think of the term with
the smallest exponent, x-1/2 as a common term.)
3. Mowing a field: A square field in a certain state park is mowed around the edges every week.
The rest of the field is kept unmowed to serve as a habitat for birds and small animals. The field measures b feet by b feet, and the mowed strip is x feet wide.
b
b
x x
x
x
3.1. Why is the area of the mowed portion b2- (b - 2 x)2?
3.2. Factor this expression to come up with an equivalent expression for the mowed area.
Homework
◼ iMath problems on section 1.3b due by Saturday, September 5. ◼ Weekly assignment 2 due Thursday, September 3.
