College Algebra: Section 1.3 (Packet #7)
By the end of the section, you should be able to:
- Classify a polynomial by number of terms and degree
- Calculate addition/subtraction/multiplication between two polynomials
- Factor polynomials using the Greatest Common Factor, Special Binomial, and by grouping
- Factor polynomials using the A-C Method
- Factor expressions containing Fractional Exponents
Definitions:
- Polynomial: algebraic expressions that:
- each term consists only of a number multiplied by a variable raised to positive integer exponents.
- Coefficient: the (big) number in any term.
- Descending Order: the powers descend from left to right.
- Leading Coefficient: the (big) number in front of the first term (in descending order)
- the (big) number in front of the term with the largest exponent.
- the (big) number in front of the term with the largest exponent.
- Constant Term: a term that consists only of a non-zero number.
- Degree of a polynomial: the largest of the degrees of the individual terms.
- Linear: polynomial of the first degree
- Quadratic: polynomial of the second degree
- Cubic: polynomial of the third degree
- Quartic: polynomial of the fourth degree
- Quintic: polynomial of the fifth degree
- (for degree 6 and larger): polynomial of the ____ degree
- Degree of the term: the sum of the exponents of the variables in a term (remember a constant has a degree of 0).
- Monomial: Polynomials consisting of a single term.
- Binomial: Polynomials consisting of two terms.
- Trinomial: Polynomials consisting of three terms.
- Factoring: reversing the process of multiplication in order to find two or more expressions whose product is the original expression.
- A Polynomial is:
- Factorable if it can be written as a product of two or more polynomials (that all have integer coefficients).
- Irreducible (over the integers) or Prime if it is not factorable.
- A Polynomial is:
Algebra of Polynomials
You can Add, Subtract, Multiply, and Divide Polynomials
You can Add, Subtract, Multiply, and Divide Polynomials
- For Addition/Subtraction: collect and combine like terms.
- For Multiplication: FOIL (First, Outer, Inner, Last) this is really the distributive property.
- For Division: cancel out variables/factors
Algebra of Polynomials Examples
🎥Examples 1&2
🎥Examples 1&2
Things to remember about the Algebra of Polynomials:
- When subtracting polynomials DO NOT FORGET to distribute the negative throughout all terms.
Factoring
There are many different methods/techniques to factoring. If one technique is not working, you may need to try another one. As you get more familiar with factoring you will start to see the patterns of when to use each technique specifically. The techniques are as followed:
There are many different methods/techniques to factoring. If one technique is not working, you may need to try another one. As you get more familiar with factoring you will start to see the patterns of when to use each technique specifically. The techniques are as followed:
- A-C Method (trinomials by grouping) for a trinomial in the form:
- Multiply a and c
- Factor a*c into two integers whose sum is (add to) b. If no such factors exist, the trinomial is irreducible over the integers
- Rewrite b in the trinomial with the sum found in Step 2, and distribute.
- The resulting polynomial of four terms will now be able to be factored by grouping.
Factoring Examples
🎥Examples 3&4 🎥Example 5 🎥Example 6
🎥Examples 3&4 🎥Example 5 🎥Example 6
Factoring Expressions that Contain FRACTIONAL exponents
These steps apply to non-polynomial expressions (expressions with fractional and negative exponents)
These steps apply to non-polynomial expressions (expressions with fractional and negative exponents)
- Identify the least exponent among the various terms
- Factor the least exponent from Step 1 from each of the terms in the expression
- Factor any other common factors from each of the terms in the expression (if applicable)
- Simplify (if applicable)
Factoring with Fractional Exponents Examples
🎥Examples 7&8
🎥Examples 7&8
Things to remember about Factoring with Fractional Exponents:
- Be careful with negative exponents when identifying the least exponent, they can be tricky/misleading.
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