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Algebra II - Polynomial Arithmetic

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    Study Guide Algebra II – Polynomial Arithme Ɵ c 1. Adding and Subtrac Ɵ ng Polynomials Polynomials are expressions made of one or more terms . Each term is separated by a plus (+) or minus ( − ) sign. In a polynomial: • The exponents on variables must be whole numbers (0, 1, 2, 3, …). • There is no limit to how long a polynomial can be. Some operations with polynomials are simple, but others require special methods. To add or subtract polynomials successfully, you need to understand: • Monomials, binomials, and trinomials • Like terms • Ascending and descending order Monomials, Binomials, and Trinomials Monomial A monomial is a single term. • a number • a variable • or a number multiplied by variables. Rules for Monomials For an expression to be a monomial: 1. Variables must have whole number exponents . 2. Variables cannot be inside square roots . 3. Variables cannot appear in denominators .

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    Study Guide Examples of Monomials • − 12 • a • 3t² • (5/8)x 2 y 3 ) • ( y 3 ) • (2(x+y 2 )/3) Expressions That Are NOT Monomials The following are not monomials : • (x - 2 ) → exponent is not a whole number • ( √ x) → variable is inside a radical • (3(x+2)/(x+3)) → variable appears in the denominator Binomial A binomial is the sum of two monomials . Example: Trinomial A trinomial is the sum of three monomials . Example:

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    Study Guide Polynomial A polynomial can be: • a monomial • the sum of two or more monomials Examples: • (5x 2 ) • (x 2 + 3x + 2) Like Terms Like terms are terms that have the same variables with the same exponents . Example of like terms: They are like terms because each contains (x 2 y) . Not Like Terms These are not like terms : Their variable parts are different. Adding Like Terms To add like terms: 1. Add the numerical coefficients. 2. Keep the variable part the same.

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    Study Guide Example: If the terms are not like terms , they cannot be combined . Example 1: Find the Sums 1. (4x 2 y + 8x 2 y) = (12x 2 y) 2. ( - 9abc + 3abc) = ( - 6abc) 3. (9xy + 7x - 28xy - 4x) Group like terms: These two terms cannot be combined further because they are unlike terms. Ascending and Descending Order When writing polynomials with one variable, we often arrange them by the size of the exponents . Descending Order The exponents go from largest to smallest . Example:

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    Study Guide Ascending Order The exponents go from smallest to largest . Example: Example 2: Write in Descending Powers of x Original expression: Arrange by powers of x : Adding Polynomials To add polynomials: 1. Combine like terms. 2. Write the result in order (usually descending order). Given expression: Step 1: Identify the Powers of (x) Arrange the terms based on the power of (x) from highest to lowest.

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    Study Guide Step 2: Arrange in Descending Order of (x) Start with the highest power of (x) and move down: Final Answer Example 3: Add the Polynomials First combine like terms. (x 3 ) terms (x 2 ) terms (x) terms

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    Study Guide constants Final answer: Vertical Method Polynomials can also be added vertically , just like numbers. Write them in descending order and line up like terms : Subtracting Polynomials To subtract polynomials, add the opposite of the polynomial being subtracted. Example 4 : Subtract from Write the expression:

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    Study Guide Change signs of the second polynomial: Combine like terms: Summary Key Ideas • A polynomial is made of one or more terms. • Terms are separated by + or − signs. Types of Expressions • Monomial: one term • Binomial: two terms • Trinomial: three terms Monomial Rules • Exponents must be whole numbers • No variables under radicals • No variables in denominators Like Terms Terms are like terms if they have: • the same variables • the same exponents Adding Like Terms • Add the coefficients • Keep the variable part the same

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    Study Guide Polynomial Order • Descending order: largest exponent → smallest • Ascending order: smallest exponent → largest Adding Polynomials 1. Combine like terms 2. Write the result in order Subtracting Polynomials • Add the opposite of the polynomial being subtracted. 2. Quiz: Adding and Subtrac Ɵ ng Polynomials 1. Question Which of the following is the monomial ? Answer Choices • (35xyz²) / 7 • (16xyz²) / (2q) • 35x + 21 Correct Answer (35xyz²) / 7 Why This Is Correct A monomial is a single algebraic term that may contain numbers and variables with non - negative exponents. ((35xyz²)/7) simplifies to 5xyz² , which is one single term. The other options are not monomials because: • ((16xyz²)/(2q)) contains a variable in the denominator. • 35x + 21 has two terms (a binomial).

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    Study Guide 2. Question Which of the following is the trinomial ? Answer Choices • (2x − 13) • (21x + 7yx² − 13) / 7 • (2x + 7y² − 13z) / 3x Correct Answer (21x + 7yx² − 13) / 7 Why This Is Correct A trinomial has exactly three terms . The expression ((21x + 7yx² − 13)/7) contains three terms in the numerator. The others are not trinomials because: • 2x − 13 has only two terms (binomial). • ((2x + 7y² − 13z)/(3x)) has variables in the denominator. 3. Question Which of the following is the binomial ? Answer Choices • 2x − 13/y • 3x² + 5x − 13 • 2x − 13

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