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4.1 Introduction to Polynomials

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Monomial: 1 term (ax n with n is a non- negative integers, a is a real number) Ex: 3x, -3, or 4xy 2 z Binomial: 2 terms Ex: 3x - 5, or 4xy 2 z + 3ab Trinomial: 3 terms Ex: 4x 2 + 2x - 3

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Polynomial: is a monomial or sum of monomials Ex: 4x 3 + 4x 2 - 2x - 3 or 5x + 2 Are these polynomials or not polynomials? No √x No| x – 3| Yes(1/2)x yesxyab yes-2 No3/xy

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Degree: exponents Degree of polynomial: highest exponent (if the term has more than 1 variable, then add all exponents of that term) Coefficient: number in front of variables Leading term: term of highest degree. Its coefficient is called the leading coefficient Constant term: the term without variable Missing term: the term that has 0 as its coefficient

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Ex: -3x 4 – 4x 2 + x – 1 Term: -3x 4, – 4x 2, x, – 1 Degree 4 2 1 0 Coefficient -3 -4 1 -1 Degree of this polynomial is 4 Leading term is -3x 4 and -3 is the leading coefficient Constant term: is -1 Missing term (s): is x 3

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Ex2: -6x 9 – 8x 6 y 4 + x 7 y + 3xy 5 - 4 Term: -6x 9, – 8x 6 y 4, x 7 y, 3xy 5, - 4 Degree 9 10 8 6 0 Coefficient -6 -8 1 3 -4 Degree of this polynomial is 10 Leading term is – 8x 6 y 4 and -8 is the leading coefficient Constant term: is -4

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Descending order: exponents decrease from left to right Ascending order: exponents increase from left to right When working with polynomials, we often use Descending order

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Arrange in descending order using power of x 1)-6x 2 – 8x 6 + x 8 + 3x - 4 = x 8 – 8x 6 - 6x 2 + 3x - 4 2)5x 2 y 2 + 4xy + 2x 3 y 4 + 9x 4 = 9x 4 + 2x 3 y 4 + 5x 2 y 2 + 4xy

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Opposites of Polynomials: 1)2x Opposite is -2x 2) 3x 4 – 4x 2 + x Opposite is - 3x 4 + 4x 2 - x

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Adding and Subtracting Polynomials Same as combining like-term: Add or subtract only numbers and keep the same variables

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1) (-6x 4 – 8x 3 + 3x - 4) + (5x 4 + x 3 + 2x 2 -7x) = -6x 4 + 5x 4 – 8x 3 + x 3 + 2x 2 + 3x -7x -4 = -x 4 - 7x 3 + 2x 2 - 4x -4

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2)(-6x 4 – 8x 3 + 3x - 4) - (5x 4 + x 3 + 2x 2 -7x) = -6x 4 – 8x 3 + 3x - 4 - 5x 4 - x 3 - 2x 2 +7x = -11x 4 - 9x 3 - 2x 2 +10x -4

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