UNIT 6 POLYNOMIALS Polynomial (Definition) A monomial or a sum of monomials. A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial. Ex. 2 x3 y4 z5 2 has a degree of 0 (do not mistake degree for zero exponent) x3 has an exponent of 3 y4 has an exponent of 4 z5 has an exponent of 5 Add up the exponents of your variables only 3 + 4 + 5 = 12 This monomial would be a 12th degree polynomial Let's Practice Finding the Degree of a Monomial 3 degree of ___ 3x degree of ___ 3x2 degree of ___ 3x3 degree of ___ 3x4 degree of ___ DEGREE OF A POLYNOMIAL The degree of a polynomial is determined by the monomial with the highest degree. 3 would be a 0 degree polynomial because all numbers have a degree of zero. (NOT TO BE CONFUSED WITH ZERO EXPONENT PROPERTY). We call 0 degree polynomials CONSTANT. 3x would be a 1st degree polynomial because the 3 has a degree of 0 and x has an exponent of 1. We call 1st degree polynomials LINEAR. 3x2 would be a 2nd degree polynomial because the 3 has a degree of 0 and x2 has an exponent of 2. We call 2nd degree polynomials QUADRATIC. 3x3 would be a 3rd degree polynomial because the 3 has a degree of 0 and x3 has an exponent of 3. We call 3rd degree polynomials CUBIC. 3x4 would be a 4th degree polynomial because the 3 has a degree of 0 and x4 has an exponent of 4. We call 4th degree polynomials QUARTIC. 3x5 would be a 5th degree polynomial because the 3 has a degree of 0 and x5 has an exponent of 5. We call 5th degree polynomials QUINTIC. 3x6 would be a 6th degree polynomial because the 3 has a degree of 0 and x6 has an exponent of 6. We call 6th degree and above their degree name, so it would be a polynomial to the 6TH DEGREE. NAMING A POLYNOMIAL The number of terms added together determines the name of a polynomial. 6 has only 1 term and would be called a MONOMIAL. x + 3 has 2 terms and would be called a BINOMIAL. x2 + x + 3 had 3 terms and would be called a TRINOMIAL. x3 + x2 + x + 3 has 4 terms and would be called a POLYNOMIAL. Any polynomial with 4 or more terms is also called a POLYNOMIAL. Use this chart as a guide Let's Practice (Fill out this table) STANDARD FORM The terms of a polynomial are usually arranged so that the terms are in order from greatest degree to least degree. This is called the standard form of a polynomial. Ex: x2 + x + 1 is in standard form because the terms are arranged from greatest to least degree from left to right. 2nd degree, 1st degree, zero degree Practice problems 7-12 on page 5 of Ch 8 Sec 1 Study Guide by writing each polynomial in standard form. Leading Coeffcient The number in front of the monomial with the largest degree. Ex. x2 + x + 1 1 is the leading coefficient since x2 is the highest degree of any of the 3 monomials. Practice problems 7-12 on page 5 of Ch 8 Sec 1 Study Guide by determining the leading coefficient of each polynomial. MULTIPLYING A POLYNOMIAL AND A MONOMIAL METHOD 1 HORIZONTAL x( 5x + x2 ) x(5x) + x(x2) distribute the x to each term inside the parenthesis 5x2 + x3 simplify x3 + 5x2 put in standard form (greatest to least degree) 1 is the leading coefficient Cubic Binomial is its proper name METHOD 2 VERTICAL x( 5x + x2 ) 5x + x2 (x) x multiply x to each term 5x2 + x3 x3 + 5x2 put in standard form (greatest to least degree) 1 is the leading coefficient Cubic Binomial is its proper name Practice multiplying by one of the methods on Worksheet 8-2 Study Guide page 12 (1-9) MULTIPLYING A BINOMIAL BY A BINOMIAL HORIZONTAL (FOIL) When 2 binomials are multiplied together, the result will always result in 4 terms. Those 4 terms will be simplified into standard form and result in a TRINOMIAL. We will concentrate on Quadratic Trinomials. Practice using the FOIL method on 8-3 Study Guide page 18 (1-18) SPECIAL PRODUCTS Ex. 1 Square of a Sum ( x + 4 )2 a is x b is 4 2(a)(b) is 2(x)(4) a2 = (x)2 2ab = 2(4x) b2 = (4)2 Put them all together to get x2 + 8x + 16 Ex. 2 Square of a difference ( x - 6 )2 a is x b is -6 2(a)(b) is 2(x)(-6) a2 = (x)2 2ab = 2(-6x) b2 = (-6)2 Put them all together to get x2 -12x + 36 Practice solving for special products on 8-3 Study Guide page 25 (1-21) Ex. 3 ( x + 4 ) ( x - 4 ) a is x b is 4 and -4 2ab is 2(x)(4) and 2(x)(-4) a2 = (x)2 2ab = 2(4x) + 2(-4x) b2 = -(4)2 Put them all together to get x2 + 8x -8x -16 Simplify x2 -16 Practice finding special products on 8-3 Study Guide page 26 (1-21) .