2 Naming Polynomials Practice: 3x4 = 4th degree monomial 5xy2= # TermsDegree1 – Monomial1 – Linear2 – Binomial2 – Quadratic3 – Trinomial3 – Cubic4+ - Polynomialth degree, etc.If a does not equal 0, the degree of axn is n.Degree of polynomials is the greatest degree of all its termsThe degree of a nonzero constant is 0.The constant 0 has no defined degree.Practice:3x4 =4th degree monomial5xy2=Cubic monomial3xy +3x +4 =Quadratic Trinomial3x4 +5xy + 6x + 2=4th degree polynomial3x2 +6x =Quadratic binomial3x3+6x2+2x =Cubic Trinomial
3 Definition of a Polynomial in x A polynomial in x is an algebraic expression of the formanxn + an-1xn-1 + an-2xn-2 + … + a1n + a0where an, an-1, an-2, …, a1 and a0 are real numbers, an ≠ 0, and n is a non-negative integer.The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.
4 Definition of a Polynomial in x A polynomial in x is an algebraic expression of the formanxn + an-1xn-1 + an-2xn-2 + … + a1n + a0where an, an-1, an-2, …, a1 and a0 are real numbers, an ≠ 0, and n is a non-negative integer.The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.Identify the 3x8 + 5x4 + 2…degree?…leading coefficient?…. and constant term?
5 Standard Form of a Polynomial Write in order of descending powers of the variable So…3x + 5x8 - 9x should be written5x8 - 9x3 +3x +10
7 Multiplying Polynomials (Ex #2) The product of two monomials is obtained by using properties of exponents. For example,(-8x6)(5x3) = -8·5x6+3 = -40x9Multiply coefficients and add exponents.Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For example,3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 + 9x4.monomialtrinomial
8 Multiplying Polynomials when Neither is a Monomial (Ex #3) Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.
9 Using the FOIL Method to Multiply Binomials lastfirst(ax + b)(cx + d) = ax · cx ax · d b · cx b · dProduct ofFirst termsProduct ofOutside termsProduct ofInside termsProduct ofLast termsinnerouter
12 The Product of the Sum and Difference of Two Terms (ex #4) DIFFERENCE OF SQUARES The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term.
13 The Square of a Binomial Sum (Ex #5) PERFECT SQUARE TRINOMIAL The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.
14 The Square of a Binomial Difference The square of a binomial difference is first term squared minus 2 times the product of the terms plus last term squared.
15 Special ProductsLet A and B represent real numbers, variables, or algebraic expressions. Special Product ExampleSum and Difference of Two Terms(A + B)(A – B) = A2 – B (2x + 3)(2x – 3) = (2x) 2 – 32= 4x2 – 9Squaring a Binomial(A + B)2 = A2 + 2AB + B (y + 5) 2 = y2 + 2·y·5 + 52= y2 + 10y + 25(A – B)2 = A2 – 2AB + B2 (3x – 4) 2 = (3x)2 – 2·3x·4 + 42= 9x2 – 24x + 16Cubing a Binomial(A + B)3 = A3 + 3A2B + 3AB2 + B (x + 4)3 = x3 + 3·x2·4 + 3·x·= x3 + 12x2 + 48x + 64(A – B)3 = A3 – 3A2B – 3AB2 + B (x – 2)3 = x3 – 3·x2·2 – 3·x·= x3 – 6x2 – 12x + 8
16 Text Example Multiply: a. (x + 4y)(3x – 5y) b. (5x + 3y) 2 Solution We will perform the multiplication in part (a) using the FOIL method. We will multiply in part (b) using the formula for the square of a binomial, (A + B) 2.a. (x + 4y)(3x – 5y) Multiply these binomials using the FOIL method.= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y)= 3x2 – 5xy + 12xy – 20y2= 3x2 + 7xy – 20y2 Combine like terms.(5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A2 + 2AB + B2= 25x2 + 30xy + 9y2FOIL