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## Polynomials

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**Monomials, binomials and trinomials are all included as**polynomials as well as any other number of terms. Polynomials means many a monomial has one term mono means one a binomial has two terms bi means two a trinomial has three terms tri means three**This is the general form for a polynomial in one variable:**The highest power on any x is called the degree of the polynomial. The coefficient on the highest powered x term is called the leading coefficient. These are constants (just fixed numbers). They are called coefficients.**Using Polynomials in an Applications**A child throws a ball upward and the height of the ball, h (in feet), can be computed by the following equations: h = -16t² + 64t + 2**a. Find the height of the ball after 0.5 sec. 1 sec, and**1.5 sec. h = -16(.5)² + 64(.5) + 2 h = -16t² + 64t + 2 30 b. Find the height of the ball at the time of release h = -16(0) + 64(0) +2 h = -16t² + 64t + 2 2**Since addition is commutative, you can reorder the terms in**a polynomial. Typically we like them in descending order which means the highest powered x term first and then the next highest etc. Variables raised to the same power are called like terms. To add two polynomials together we just combine like terms. add these**Polynomials can also be added by combining like terms in**columns. The sum of the polynomials is shown here Place holders such as 0 and 0c may be used to help line up like terms. To add two polynomials together we just combine like terms. add these**To subtract these two polynomials we just make sure to get**the negative sign to each term we are subtracting and then we combine like terms.**To subtract these two polynomials we just make sure to get**the negative sign to each term we are subtracting and then we combine like terms. Group like terms Combine like terms**Subtracting Polynomials**Subtract from , and simplify the result Apply the distributive property. Group like terms in descending order. The t²-terms are the only like terms. Get a common denominator for the t²-terms. Add Like terms.**Multiplying Monomials**(3x⁴)(4x²) Group coefficients and like bases (3∙4)(x⁴∙x²) Add the exponents and simply Group coefficients and like bases. Simplify**Multiplying a polynomial by a Monomials**Multiply each term of the polynomial by 2t. 2t(4t - 3) Apply the distributive property. (2t)(4t) + (2t)(-3) 8t² - 6t Simplify each term. Multiply each term of the polynomial by -3a² -3a² (-4a² + 2a - ⅓) Apply the distributive property (-3a²)(-4a²) + (-3a²)(2a) + (-3a²)(-⅓) 12a⁴ -6a³ + a² Simplify each term.**MULTIPLYING POLYNOMIALS**When multiplying polynomials you need to multiply each term in the first polynomial by each term in the second polynomial. A mnemonic to help us accomplish this when multiplying two binomials is: F O I L AST AST IRST IRST nner nner UTER UTER**If you do not have two binomials, you need to make sure that**each term from the first polynomial is multiplied to each term of the second polynomial. If you line up like terms as you are multiplying it makes this step easier. We'll then need to combine like terms.**Multiplication of polynomials can be performed vertically by**a process similar to column multiplication of real numbers. Line up the polynomials similar to column multiplication of real numbers. When multiplying by the column method, it is important to align like terms vertically before adding terms. 3x² - 2x + 4 2x - 3 Add like terms**Special Case Products:Difference of SquaresPerfect Square**Trinomials Special Case Product Formulas (a + b)(a – b) = a² - b² The product is called a difference of squares. (a + b)² = a² + 2ab + b² The product is called a perfect (a – b)² = a² - 2ab + b² square trinomial.**1. The first special case occurs when multiplying the sum**and difference of the same two terms. FOIL Notice that the middle terms are opposites. This leaves only the difference between the square of the first term and the square of the second term. For this reason, the product is called a difference of squares. (2x + 3)(2x – 3) 4x² -6x + 6x - 9 Combine Like terms 4x² - 9**2. The second special case involves the square of a**binomial. When squaring a binomial, the product will be trinomial called a perfect square trinomial. The first and third terms are formed by squaring the terms of the binomial. The middle term equals twice the product of the terms in the binomial. FOIL (3x + 7)(3x + 7) Combine Like terms 9x² +21x + 21x + 49 9x² + 42x + 49 (3x)² + 2(3x)(7) + (7)² The expression (3x – 7)² also expands to a perfect square trinomial, but the middle term will be negative: (3x – 7)(3x – 7)= 9x² -21x -21x + 49 =9x² -42x +49