Rational Expressions PreRequisite Skills: Exponents and Factoring

The skills developed in this lesson are included to support the work you will do in this unit. Exponent Law manipulation and Factoring are critical skills when working with Rational Expressions.

Information A power consists of a base and an exponent. Powers can be multiplied by a constant called a coefficient. Exponent laws are rules that help to simplify or evaluate powers.

Example 1 Find the following information given the expression 3x 2. coefficient power exponent variable base Identify the parts of a monomial 3 x2x2 2 x x It’s important to note that the exponent only applies to the x.

Example 2 Use exponent laws to simplify or evaluate the following expressions a) b) Using the Exponent Laws When multiplying powers of the same base, we add the exponents. When dividing powers of the same base, we subtract the exponents.

Example 2 c) d) Using the Exponent Laws When dividing powers of the same base, we subtract the exponents.

More Information

Example 3 Simplify the following. Express your answer in descending order. a) Simplifying Polynomials Group all ‘like terms’ together. Then combine them. Remember: like terms are those that have the same variables with the same exponents.

Example 3 Simplify the following. Express your answer in descending order. b) Simplifying Polynomials If there are any brackets, we must remove these first. If they are separated by an addition sign, you can simply remove the brackets. Now continue like before.

Example 3 Simplify the following. Express your answer in descending order. c) Simplifying Polynomials If there are any brackets, we must remove these first. If they are separated by a subtraction sign, you have to apply the negative to all terms in the brackets that follow it. Now continue like before.

Example 3 Simplify the following. Express your answer in descending order. d) Simplifying Polynomials If there are any brackets, we must remove these first. If a number proceeds the second set of brackets, you have to apply the number (multiply it in) to all terms in the brackets that follow it.

Example 3 Simplify the following. Express your answer in descending order. e) Simplifying Polynomials If there two sets of brackets that are multiplied together, we remove these by multiplying using FOIL. OILF F – multiply the first terms together O – multiply the outside terms together I – multiply the inside terms together L – multiply the last terms together

Example 3 Simplify the following. Express your answer in descending order. f) Simplifying Polynomials If there two sets of brackets that are multiplied together, we remove these by multiplying using FOIL. OILF F – multiply the first terms together O – multiply the outside terms together I – multiply the inside terms together L – multiply the last terms together

Check the factors for another difference of squares. Check for a difference of squares. Check if the trinomial is of the form Factor using any method: sum & product, box method, decomposition, guess & check. 2 terms3 terms Is there a common factor? Factor out the greatest common factor. How many terms are there? yesno This factoring flowchart can be used to guide you as you factor a polynomial. We don’t deal with 3 term factoring until later in the unit…

Example 4 Factor the greatest common factor, if possible. Factoring out the Greatest Common Factor a) b) c) Is there a common factor? If so, factor it out.

Example 4 Factor the following polynomial expressions, if possible. Factoring a Difference of Squares a) b) Is there a common factor in either of these? Two terms. Factor the difference of squares.

Example 4 Factor the following polynomial expressions, if possible. Factoring a Difference of Squares c) Is there a common factor? Factor this out first! Two terms. Factor the difference of squares!

Need to Know A power consists of a base and an exponent. can be multiplied by a number called a coefficient. can be multiplied with or divided by other powers. can be added to or subtracted from other powers.

Need to Know Exponent laws are rules that help to simplify or evaluate powers. Some of these laws are in the table below. Rule Explanation Product Rule  when multiplying powers with the same base, add the exponents Quotient Rule  when dividing powers with the same base, subtract the exponents Zero Power Rule  any power with an exponent of 0 equals 1 Negative Exponent Rule  A power with a negative exponent can be written as the reciprocal of the power with a positive exponent.

Need to Know Sometimes when powers are added or subtracted together, a polynomial expression is created. A polynomial expression contains: real number coefficients whole number exponents can be written in expanded form or factored form

Need to Know To factor any polynomial expression: Check for a common factor If there are 2 terms, try to factor using a difference of squares If there are 3 terms, try to factor using sum & product, box method, or decomposition. You’re ready! Try the homework from this section.