Elementary Algebra Review

Factoring Trinomials

Quadratic trinomials factor into the product of two binomials . Factoring trinomials is based on the FOIL
method of multiplication. Note that the first term of the trinomial is the product of the first terms of the
binomials. The last term of the trinomial is the product of the last terms of the binomials. The middle
term of the trinomial is the sum of the products of the two inner terms and the two outer terms.

To factor a quadratic trinomial, first list all possible factors for the first term and the last term. Write out
these choices as trial factors; then find the sum of the outer and inner products. Compare this sum to the
middle term to determine the correct factorization.

Example 1: Factor X² – 7X + 12
The last term has a positive sign; therefore both signs are alike . The sign on the
middle term is negative; therefore both signs are negative. The factors of X² must
be X and X. The factors of 12 are -1, -12 or -2, -6 or -3, -4.
 

Trial factors	Sum outer and inner products

Therefore the answer is (X – 3)(X – 4)

Example 2: Factor 4A² – A – 5

The last term has a negative sign; therefore one sign will be positive, and the other will be
negative. The factors of 4A² are A and 4A or 2A and 2A. The factors of -5 are -1 and 5 or 1 and
-5.

Trial factors	Sum of outer and inner products

Therefore the answer is (A + 1)(4A – 5)

Simple Quadratic Equations

To solve quadratic equations use the Principle of Zero Products, which states that if the product of two
factors is zero, then at least one of the factors must be zero.

Example: Solve
		Put into standard form
		Factor
		Use the Principle of Zero Products
		Solve each equation

The solutions are and 1.

Exercises

Factor each of the following completely

Solve each of the following questions

9. Find three consecutive positive odd integers such that twice the product of the first two minus the
product of the first and third is 49.

10. The length of a rectangle is 3ft. more than its width. Its area is 28 square ft. Find its dimensions.

11. A room contains 54 chairs. The number of chairs per row is three less than twice the number of rows.
Find the number of rows and the number of chairs per row.

12. A strip of uniform width is to be cut off of both sides and both ends of a sheet of paper that is 8
inches by 11 inches in order to reduce the size of the paper to an area of 40 square inches. Find the width
of the strip.

13. A brace wire is attached to the top of a tower 24m tall. It is anchored 7m from the base of the tower.
How long is the piece of wire between these two points?

14. A ladder is leaning against a wall. The vertical distance up the wall to the top of the ladder is 4ft. less
than the length of the ladder. The distance from the base of the ladder to the wall is 8ft. less than the
length of the ladder. Find the length of the ladder.

RATIONAL EXPRESSIONS

Multiplying and Dividing Rational Expressions

Rational expressions are fractions where the numerator or the denominator or both are polynomials . The
product of two rational expressions is a rational expression whose numerator is the product of the
numerators and whose denominator is the product of the denominators.

Example:
		Factor
		Multiply
		Simplify
		Write answer in simplest form

The quotient of two rational expressions is the product of the first rational expression and the reciprocal of
the second rational expression.
 

Example:
		Write as a multiplication problem
		Factor, Multiply, and Simplify
		Write answer in simplest form

Adding and Subtracting Rational Expressions

When adding or subtracting rational expressions with the same denominator, add or subtract the
numerators and keep the common denominator. When adding or subtracting rational expressions with
different denominators, first find the least common denominator for the expressions. Next express each
rational expression in terms of the common denominator. Finally add or subtract the rational expressions
and simplify the result.

Example:
		The denominators are factored

The least common denominator is (Y – 1)(Y – 4)

	Express each using the LCD
	Add the rational expressions
	Simplify
	There are no common factors.

Complex Fractions

When both the numerator and the denominator of a complex fraction contain a single fraction, rewrite the
complex fraction as a division problem and do the indicated division. If the numerator or the denominator
contain more than one term, the terms must be combined before doing the division.

Example:

Working the numerator first

	Rewrite using the LCD
	Add
	Simplify

Working the denominator next

	Rewrite using the LCD
	Subtract
	Simplify

Thus

	Write as a division problem
	Write as equivalent multiplication problem
	Multiply and Simplify

Solving Equations with Rational Expressions

Equation containing rational expressions should first be cleared of all denominators by multiplying every
term in the equation by the LCM of the denominators . The resulting linear or quadratic equation is then
solved in the usual way. Always check all answers to eliminate any candidates that result in division by
zero (excluded values).

Example:
		Factor the denominators

The LCM of the denominators is (X – 6)(X + 6) and then multiply by the LCM

Resulting Equation Simplify

Since the excluded values are 6 and -6, 15 is the solution to the equation.