Polynomials

The Degree of ax n If a does not equal 0, the degree of ax n is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.

Definition of a Polynomial in x A polynomial in x is an algebraic expression of the form a n x n + a n-1 x n-1 + a n-2 x n-2 + … + a 1 n + a 0 where a n, a n-1, a n-2, …, a 1 and a 0 are real numbers. a n = 0, and n is a non-negative integer. The polynomial is of degree n, a n is the leading coefficient, and a 0 is the constant term.

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

The product of two monomials is obtained by using properties of exponents. For example, (-8x 6 )(5x 3 ) = -8·5x 6+3 = -40x 9 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, 3x 4 (2x 3 – 7x + 3) = 3x 4 · 2x 3 – 3x 4 · 7x + 3x 4 · 3 = 6x 7 – 21x 5 + 9x 4. monomialtrinomial Multiplying Polynomials

Multiplying Polynomials when Neither is a Monomial Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.

Using the FOIL Method to Multiply Binomials (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 first last inner outer

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

Multiply: (3x + 4)(5x – 3). Solution (3x + 4)(5x – 3)= 3x·5x + 3x(-3) + 4(5x) + 4(-3) = 15x 2 – 9x + 20x – 12 = 15x x – 12 Combine like terms. first last inner outer FOIL Text Example

The Product of the Sum and Difference of Two Terms 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.

The Square of a Binomial Sum The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.

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.

Let A and B represent real numbers, variables, or algebraic expressions. Special Product Example Sum and Difference of Two Terms (A + B)(A – B) = A 2 – B 2 (2x + 3)(2x – 3) = (2x) 2 – 3 2 = 4x 2 – 9 Squaring a Binomial (A + B) 2 = A 2 + 2AB + B 2 (y + 5) 2 = y 2 + 2·y· = y y + 25 (A – B) 2 = A 2 – 2AB + B 2 (3x – 4) 2 = (3x) 2 – 2·3x· = 9x 2 – 24x + 16 Cubing a Binomial (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3 (x + 4) 3 = x 3 + 3·x 2 ·4 + 3·x· = x x x + 64 (A – B) 3 = A 3 – 3A 2 B – 3AB 2 + B 3 (x – 2) 3 = x 3 – 3·x 2 ·2 – 3·x· = x 3 – 6x 2 – 12x + 8 Special Products

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) = 3x 2 – 5xy + 12xy – 20y 2 = 3x 2 + 7xy – 20y 2 Combine like terms. (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A 2 + 2AB + B 2 = 25x xy + 9y 2 FOIL Text Example

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

Polynomials