Week 5 Day 1 (Factoring Linear Equations, Algebraic Fractions, Graphing on Number Lines, Absolute Value)

Week 5, Day 1

 

Factoring p. 122

Factoring polynomials requires finding the algebraic expressions whose product equals the original polynomial.

 

Factoring a Common Monomial Factor

Here we find the two (or more) quantities whose product equals the desired quantity.

First, find the largest common monomial factor of each term.

Then, divide the original polynomial by the monomial factor to obtain the second factor (which will usually be a polynomial).

 

Factoring the Difference of Two Squares p. 123

a² – b² = (a – b)(a + b)

First, find the square root of each term.

Then express your answer as the product of the sum of those roots times the difference of those roots.

 

Factoring Three Terms in the form Ax² ± Bx ± C p. 127

1.     Check to see if you can monomial factor. If A = 1, use double parentheses and factor these terms on the left side of each parentheses.

2.     Factor the last term to the right side.

3.     Decide on the signs (+ or –):

a.      If the sign of C is negative

                                                    i.     Find two numbers whose product is C and whose difference is B.

                                                  ii.     Give the larger of those numbers the sign of B, and the opposite sign to the smaller number.

b.     If the sign of C is positive

                                                    i.     Find two numbers whose product is C and whose sum is B.

                                                  ii.     Give both numbers the sign of B.

4.     Now check your work by using FOIL to compute the resulting polynomial.

 

Solving Quadratic Equations p. 130

For Ax² + Bx + C = 0 (when A ¹ 0)

1.     Put all terms in descending order to one side of the equal sign and leave zero opposite.

2.     Factor.

3.     Set each factor equal to zero.

4.     Solve each factor.

5.     Check your answers by plugging them into the original equation.

 

Algebraic Fractions p. 132

These have at least one variable in the numerator or the denominator. We must be very careful to NEVER let the denominator equal zero.

Reduce algebraic fractions to their lowest terms by factoring the numerator and/or the denominator and then canceling the common factors where possible.

 

Multiplying Algebraic Fractions p. 133

First, factor any polynomials and cancel factors where possible. Then multiply numerators to numerators and denominators to denominators.

 

Dividing Algebraic Fractions p. 134

This is very much like dividing numerical fractions: just invert the divisor and multiply – but don’t cancel until after you have inverted the divisor!

Add & Subtract Algebraic Fractions p. 135

If they have the same denominator, you can add or subtract the numerators and reduce if possible. If they have different denominators, you must first find the lowest common denominator, change them to both to equivalent fractions, and then proceed as usual.

 

START HERE WEDNESDAY

Inequalities p. 139

Inequalities use the following signs:

x < y ß “x is less than y”

x £ y ß “x is less than or equal to y”

x ³ y ß “x is greater than or equal to y”

x > y ß “x is greater than y”

 

Treat inequalities like regular equations EXCEPT when multiplying or dividing by a negative number – in this case, you must reverse the direction of the inequality sign.

Graphing on Number Lines p. 140

Earlier we saw where numbers were positioned on number lines; now we will look at how to graph inequalities on number lines, and later we will see how two number lines come together to create a coordinate graphing system known as the Cartesian plane.

·       Integers alone use single dots

·       Inequalities use lines, line segments, rays, dots, and hollow dots

o   Lines indicate infinite values

o   Line segments indicate limited or finite values

o   Rays indicate infinite values in one direction

§  A ray with a dot is a closed ray or closed half line

§  A ray with a hollow dot is an open ray or open half line

o   Dots indicate that a value is included

o   Hollow dots indicate that a value is excluded

·       Intervals use line segments with dots or hollow dots

o   An interval with only dots is a closed interval or inclusive interval

o   An interval with only hollow dots is an open interval or exclusive interval

o   An interval with one dot and one hollow dot is a half-open interval or partial interval

Examples: see p. 140-142

 

Absolute Value p. 142

Absolute value measures an expression’s distance from zero and is written as |x| or |-3| or |x + 2|, etc. Absolute value is always positive except for the absolute value of zero, which equals zero. Note that the answer to example 7 is “no solution” because no absolute value can be negative, and the answer to 8 is “all real numbers” because any value will result in a positive number in this expression.