Operations with Polynomials

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Operations with Polynomials:

Operations with P o l y n o m i a l s

Polynomial means “many terms”:

Polynomial means “many terms” Monomial: a single term Ex: 3x, 8, 9y² Binomial: TWO terms Ex: 5x + 8, x - 7 Trinomial: THREE terms Ex: x² + 7x + 12

Degree of a Polynomial:

Degree of a Polynomial Degree: degree of the term with the greatest exponents Example: Find the degree of : 5x³ + 2x² - 4x + 8 The degree of the polynomial is 3

Simplifying Polynomials:

Simplifying Polynomials Simplify by combining like terms: 3mn – 12m² + 6mn + 3m 2 – 7mn – 9m 2 + 2mn 3x + 4y – 9 – 6x – 10y + 6 – x + y -4x + y - 3

Adding Polynomials:

Adding Polynomials To add polynomials, combine like terms. (5x² + 2xy) + (-3x² - 7xy) 2x² - 5xy Try: (-6x + 2xy – y) + (2x – 2xy + 4y) -4x + 3y

Subtracting Polynomials:

Subtracting Polynomials Change subtraction to addition… ADD THE OPPOSITE! (8x + 3) - (4x – 6) 8x + 3 – 4x + 6 4x + 9 Try : (7x² - 2x + 4) – (-3x² + 8x + 1) 7x² - 2x + 4 + 3x² - 8x – 1 10x² - 10x +3

Multiplying a Monomial times a Binomial:

Multiplying a Monomial times a Binomial –6a 4 b – 10a 2 b 3 + 4a 2 b Try : 9x ( 2x² - 5x + 8) 18x³ - 45x² + 72x 2ab( -3a³ - 5ab² + 2a) Use the Distributive Property!

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(x + 5)(x – 3) F O I L x 2 – 3x + 5x – 15 x 2 + 2x – 15 Multiplying Binomials Try : (x - 4)(x - 7) x ² – 7x - 4x – 28 x ² - 11x – 28

Squaring Binomials:

Squaring Binomials (x + 4) 2 (x + 4)(x + 4) x 2 +4x + 4x + 16 x 2 + 8x + 16 Here’s a shortcut... (a + b) 2 a 2 + 2ab + b 2

Multiply these binomials:

Multiply these binomials (x - 4)(x + 4) x 2 + 4x - 4x + 16 x 2 – 16 Here’s a shortcut: (a + b)(a – b) a 2 - b 2