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Published byRosamund Boone Modified over 8 years ago

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A polynomial is an algebraic expression that includes addition, subtraction, multiplication, and whole number exponents, such as: 4x 3 – 3x 2 + 7x + 5

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A polynomial can not include roots or fractional exponents fractions with the variable on the bottom (or negative exponents) variables in the exponent

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“Polynomial” means “many parts”. The parts of a polynomial are called terms. Each + or – starts a new term. 4x 3 – 3x 2 + 7x + 5 has four terms.

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How many terms are in each of these polynomials? x 2 – 2x + 3 4n 5 + 2n 4 + 3n 3 – 2n 2 5x 2 y 5 + 2x 3 y 6

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How many terms are in each of these polynomials? x 2 – 2x + 3 3 4n 5 + 2n 4 + 3n 3 – 2n 2 4 5x 2 y 5 + 2x 3 y 6 2

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In the polynomial 4x 3 + 2y 2 – 5z + 2 How many terms are there? What is the coefficient on the 1 st term? What is the coefficient on the 3 rd term? What is the base on the 2 nd term? Which term is a constant?

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In the polynomial 4x 3 + 2y 2 – 5z + 2 How many terms are there? 4 What is the coefficient on the 1 st term? 4 What is the coefficient on the 3 rd term? -5 What is the base on the 2 nd term? y Which term is a constant? 4 th

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We often classify polynomials by the number of terms they have. In particular … Monomial Binomial Trinomial

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We often classify polynomials by the number of terms they have. In particular … Monomial=1 term Binomial =2 terms Trinomial=3 terms

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Your book will sometimes ask you to write polynomials in standard form. This just means re-arranging the terms so the exponents count down from left to right.

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You can have more than one variable in a polynomial. For instance 5x 2 y 4 + 2xy 3 is a binomial where each term has 2 variables.

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The degree of a term tells how big that term is. The degree is the sum of the exponents in a term. In 5x 2 y 4 + 2xy 3, the degree of the 1 st term is 6 and the degree of the 2 nd term is 4.

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What is the degree of each term of this polynomial? 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x + 5

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What is the degree of each term of this polynomial? 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x + 5 4 5 10 1 0

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The degree of a whole polynomial is the largest of the degrees of its terms. So, the degree of 5x 2 y 4 + 2xy 3 is 6. The degree of 5x 2 y 2 – 2x 5 + 4x 7 y 3 – 7x + 5 is 10.

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What is the degree of 5n 5 p 3 + 2n 2 p 7 – 4n 4 p 3 + 5np 6 ?

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What is the degree of 5n 5 p 3 + 2n 2 p 7 – 4n 4 p 3 + 5np 6 ? 8 9 7 7 The overall degree is 9.

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Special types of polynomials: Quadratic One variable Degree = 2 Cubic One variable Degree = 3

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It’s easy to add or subtract polynomials. Just combine like terms. When you do that, you add the coefficients, and leave the variables and coefficients alone.

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(5x 2 + 3x – 2) + (3x 2 – 7x + 8) =8x 2 – 4x + 6

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The variables and exponents act like a label for the terms, which is why they don’t change when you add or subtract.

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Simplify: (6x 2 y 4 + 2x 3 y + 3x 2 y 2 ) + (5x 3 y – 2x 2 y 4 + 2x 2 y 2 )

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Simplify: (6x 2 y 4 + 2x 3 y + 3x 2 y 2 ) + (5x 3 y – 2x 2 y 4 + 2x 2 y 2 ) = 4x 2 y 4 + 7x 3 y + 5x 2 y 2

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Simplify: (y 3 – 2y 2 + 3y + 1) + (3y 3 + 2y 2 – 5y + 2)

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Simplify: (y 3 – 2y 2 + 3y + 1) + (3y 3 + 2y 2 – 5y + 2) 4y 3 – 2y + 3

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Simplify: (7n 4 + 8n 3 – 5n 2 ) – (2n 4 + 3n 3 + n 2 )

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Simplify: (7n 4 + 8n 3 – 5n 2 ) – (2n 4 + 3n 3 + n 2 ) 5n 4 + 5n 3 – 6n 2

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Simplify: (5x 2 – 3x + 5) – (2x 2 + 4x – 1)

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Simplify: (5x 2 – 3x + 5) – (2x 2 + 4x – 1) 3x 2 – 7x + 6

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Simplify: (7x 2 + 2y 2 ) – (4x 2 + 2xy – 3y 2 )

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Simplify: (7x 2 + 2y 2 ) – (4x 2 + 2xy – 3y 2 ) 3x 2 – 2xy + 5y 2

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