ALGEBRA 1 UNIT 8 POLYNOMIAL EXPRESSIONS (See Part 2 for Factoring) Unit Essential Questions   Are two algebraic expressions that appear to be different actually equivalent? What is the relationship between properties of real numbers and properties of polynomials?

ADDING AND SUBTRACTING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.

WARM UP Simplify. 1) 2) 3) 8x + 2y h + 5j –4a + 5b

KEY CONCEPTS AND VOCABULARY A monomial is a real number, a variable, or the product of real numbers and variables (Note: the variables must have positive integer exponents to be a monomial). The degree of a monomial is the sum of the exponents of its variables. A polynomial is a monomial or a sum of monomials. Standard form of a polynomial means that the degrees of its monomial terms are written in descending order. The degree of a polynomial is the same as the degree of the monomial with the greatest exponent.

KEY CONCEPTS AND VOCABULARY CLASSIFICATION OF POLYNOMIALS DEGREE NUMBER OF TERMS Constant 1 Monomial Linear 2 Binomial Quadratic 3 Trinomial Cubic 4 Polynomial with 4 terms EXAMPLES OF MONOMIALS EXAMPLES OF NOT 6 g

EXAMPLE 1: IDENTIFYING POLYNOMIALS Determine whether each expression is a polynomial. If it is a polynomial, classify the polynomial by the degree and number of terms. a) b) c) d) Polynomial, Cubic Trinomial Not a Polynomial 5 Polynomial, Quadratic Binomial Polynomial, Constant Monomial

EXAMPLE 2: WRITING POLYNOMIALS IN STANDARD FORM Write the polynomial in standard form. Then identify the leading coefficient. a) b) c) Leading Coefficient = 3 Leading Coefficient = –2 Leading Coefficient = 8

EXAMPLE 3: ADDING POLYNOMIALS Simplify. a) b)

EXAMPLE 4: SUBTRACTING POLYNOMIALS Simplify. a) b)

EXAMPLE 5: SIMPLIFYING USING GEOMETRIC FORMULAS Express the perimeter as a polynomial. a) b)

EXAMPLE 6: ADDING AND SUBTRACTING POLYNOMIALS IN REAL-WORLD APPLICATIONS The equations and represent the number of Miami Heat hats, H, and the number of Cleveland Cavalier hats, C, sold in m months at a sports store. Write an equation for the total, T, of Heat and Cavalier hats sold. Predict the number of Heat and Cavalier hats sold in 9 months. 267 Hats

RATE YOUR UNDERSTANDING ADDING AND SUBTRACTING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients. RATING LEARNING SCALE 4 I am able to add and subtract polynomials in real-world applications or in more challenging problems that I have never previously attempted 3 identify a polynomial and write polynomials in standard form add and subtract polynomials 2 identify a polynomial and write polynomials in standard form with help add and subtract polynomials with help 1 identify the degree of a monomial   TARGET

MULTIPLYING A POLYNOMIAL BY A MONOMIAL MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

WARM UP Simplify. 1) 2) 3)

KEY CONCEPTS AND VOCABULARY You can use the Distribution Property to multiply a monomial by a polynomial.

EXAMPLE 1: MULTIPLYING A POLYNOMIAL BY A MONOMIAL Simplify. a) b)

EXAMPLE 2: SIMPLIFYING EXPRESSIONS WITH A PRODUCT OF A POLYNOMIAL AND A MONOMIAL Simplify. a) b)

EXAMPLE 3: SIMPLIFYING USING GEOMETRIC FORMULAS Express the area as a polynomial. a) b)

EXAMPLE 4: SOLVING EQUATIONS WITH POLYNOMIALS ON EACH SIDE Solve. a) b) x = –1

RATE YOUR UNDERSTANDING MULTIPLYING A POLYNOMIAL BY A MONOMIAL MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. RATING LEARNING SCALE 4 I am able to multiply a polynomial by a monomial in more challenging problems that I have never previously attempted (such as solving equations) 3 multiply a polynomial by a monomial 2 multiply a polynomial by a monomial with help 1 understand that the distributive property can be applied to polynomials   TARGET

MULTIPLYING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

WARM UP Simplify. 1) 2) 3)

KEY CONCEPTS AND VOCABULARY METHODS FOR MULTIPLYING POLYNOMIALS DISTRIBUTIVE PROPERTY METHOD FOIL METHOD Example:  

EXAMPLE 1: FINDING THE PRODUCT OF TWO BINOMIALS USING THE DISTRIBUTIVE PROPERTY Simplify using the distributive property. a) b) c)

EXAMPLE 2: FINDING THE PRODUCT OF TWO BINOMIALS USING THE FOIL METHOD Simplify using the FOIL method. a) b) c)

EXAMPLE 3: FINDING THE PRODUCT OF A BINOMIAL AND TRINOMIAL Simplify using the distributive property. a) b) c)

EXAMPLE 4: SIMPLIFYING PRODUCTS Simplify. a) b)

RATE YOUR UNDERSTANDING MULTIPLYING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. RATING LEARNING SCALE 4 I am able to multiply two binomials or a binomial by a trinomial in more challenging problems that I have never previously attempted 3 multiply two binomials or a binomial by a trinomial 2 multiply two binomials or a binomial by a trinomial with help 1 understand the distributive property   TARGET

SPECIAL PRODUCTS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

WARM UP Simplify. 1) 2) 3)

KEY CONCEPTS AND VOCABULARY MULTIPLYING SPECIAL CASES THE SQUARE OF A BINOMIAL THE PRODUCT OF A SUM AND DIFFERENCE (a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 Or (a – b)2 = (a – b)(a – b) = a2 – 2ab + b2 (a + b)(a – b) = a2 – b2

EXAMPLE 1: SIMPLIFYING THE SQUARE OF A BINOMIAL (SUM) Simplify. a) b) c)

EXAMPLE 2: SIMPLIFYING THE SQUARE OF A BINOMIAL (DIFFERENCE) Simplify. a) b) c)

EXAMPLE 3: SIMPLIFYING THE PRODUCT OF A SUM AND DIFFERENCE Simplify. a) b) c) d)

EXAMPLE 4: SIMPLIFYING MORE CHALLENGING PROBLEMS WITH SPECIAL CASES Simplify. a) b) c) d)

EXAMPLE 4: SIMPLIFYING MORE CHALLENGING PROBLEMS WITH SPECIAL CASES Simplify. e) f)

RATE YOUR UNDERSTANDING SPECIAL PRODUCTS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. RATING LEARNING SCALE 4 I am able to simplify special products in more challenging problems that I have never previously attempted 3 find the square of a binomial find the product of a sum and difference 2 find the square of a binomial with help find the product of a sum and difference with help 1 understand that there are special rules to simplify the square of a binomial and the product of a sum and difference   TARGET