Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest Common Factor (GCF) VIII. Factoring Trinomials IX. Solving Quadratic Equations by Factoring 1
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can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. coefficient of a term is the numerical part. polynomial is a finite sum of terms in which all variables are raised to nonneg powers & no variables appear in denominators. Which of the following are polynomials? 3
Monomial Remark: We do not have any special names for polynomials with four or more terms (other than just calling it a polynomial) degree of a polynomial degree of the largest term. Identify the terms and their degree. What is the degree of the polynomial? Term Degree of term Degree of polynomial = 4
leading term of a polynomial is the term of highest degree. Its coefficient is leading coefficient. Polynomial Number of terms Type Degree of polynomial Leading Term Leading Coefficient 10 or trinomial where appropriate. 5
Generally we write the polynomials in descending order, that is from the highest degree term down to the lowest. This is called Standard Form of the polynomial. when x = -3. 6
Like terms are monomials (single terms) containing exactly same variable(s) to same powers (exponents). If two terms are like, then we can add or subtract them. 7
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Exercise: 1. Simplify: 2. Find the opposite polynomial to 3. Subtract: 4. Subtract from 11
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ft. Find a polynomial for the remaining area of the deck. 13
VI. Multiplication of Polynomials Case 1: Product of Monomials 1) (5x)(3x) 14
Case 2: Monomial x Polynomial 2) x(x 3-5) 3) 4x 2 (x - 5) 15
4) -4x(x - 5) 16
Note the differences between addition/subtraction vs. multiplication To add or subtract, the terms must be. 17
Case 3: Binomial x Binomial = FOIL Method The FOIL Method O F (x - 6)(x + 2) I L FOIL First Outside Inside Last = F + O + I + L = = 18
1) (x - 3)(x + 2) 2) (x - 5)(x - 7) 3) (3x - 4)(2x + 5) 19
Quick Check! (2x - 5) + (x + 2) (2x - 5)(x + 2) 20
Case 4: Polynomial x Polynomial 21
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Homework Chapter 4 Part I: Exponents & Their Properties Simplify the following Expressions 23
Part II: Negative Exponents Simplify the following expressions. Your final answer should NOT include negative exponents. Part III: Scientific Notation Multiply or divide as appropriate. Write final answer in scientific notation. 24
Part IV: Polynomials For each polynomial below (a) classify it as a monomial, binomial, trinomial, or a polynomial with no special name; (b) give the degree of the polynomial; (c) determine the leading term and leading coefficient. Part V: Addition & Subtraction of Polynomials Add or Subtract as appropriate 25
Part VI: Multiplication of Polynomials Multiply the following polynomials 26
Part VII: Greatest Common Factor (GCF) Factor out the GCF from each of the following polynomials Factor the following polynomials by grouping (if possible) 27
Part VIII: Factoring Trinomials Factor the following trinomials. If it is not possible to factor the trinomial then state that it is prime 28
Part IX: Solving Quadratic Equations by Factoring Solve the following equations by factoring 29