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Polynomials P4

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**Naming Polynomials Practice: 3x4 = 4th degree monomial 5xy2=**

# Terms Degree 1 – Monomial 1 – Linear 2 – Binomial 2 – Quadratic 3 – Trinomial 3 – Cubic 4+ - Polynomial th degree, etc. If a does not equal 0, the degree of axn is n. Degree of polynomials is the greatest degree of all its terms The degree of a nonzero constant is 0. The constant 0 has no defined degree. Practice: 3x4 = 4th degree monomial 5xy2= Cubic monomial 3xy +3x +4 = Quadratic Trinomial 3x4 +5xy + 6x + 2= 4th degree polynomial 3x2 +6x = Quadratic binomial 3x3+6x2+2x = Cubic Trinomial

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**Definition of a Polynomial in x**

A polynomial in x is an algebraic expression of the form anxn + an-1xn-1 + an-2xn-2 + … + a1n + a0 where 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.

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**Definition of a Polynomial in x**

A polynomial in x is an algebraic expression of the form anxn + an-1xn-1 + an-2xn-2 + … + a1n + a0 where 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?

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**Standard Form of a Polynomial**

Write in order of descending powers of the variable So…3x + 5x8 - 9x should be written 5x8 - 9x3 +3x +10

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**Adding and Subtracting Polynomials (Ex#1)**

Perform the indicated operations and simplify: (-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6) Solution (-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6) = (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms. = 4x3 + 9x2 – (-13x) + (-3) Combine like terms. = 4x3 + 9x2 + 13x – 3

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**Multiplying Polynomials (Ex #2)**

The product of two monomials is obtained by using properties of exponents. For example, (-8x6)(5x3) = -8·5x6+3 = -40x9 Multiply 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. monomial trinomial

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**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.

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**Using the FOIL Method to Multiply Binomials**

last first (ax + b)(cx + d) = ax · cx ax · d b · cx b · d Product of First terms Product of Outside terms Product of Inside terms Product of Last terms inner outer

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Ex #3 Multiply: (3x + 4)(5x – 3).

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**Text Example Multiply: (3x + 4)(5x – 3). Solution **

(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3) = 15x2 – 9x + 20x – 12 = 15x2 + 11x – 12 Combine like terms. last first F O I L inner outer

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**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.

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**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.

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**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.

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Special Products Let A and B represent real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Two Terms (A + B)(A – B) = A2 – B (2x + 3)(2x – 3) = (2x) 2 – 32 = 4x2 – 9 Squaring 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 + 16 Cubing 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

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**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 + 9y2 F O I L

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**Example ( 3x + 4 )2 =(3x)2 + (2)(3x) (4) + 42 =9x2 + 24x + 16**

Multiply: (3x + 4)2. Solution: ( 3x + 4 )2 =(3x)2 + (2)(3x) (4) =9x2 + 24x + 16

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Polynomials

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