Polynomials and Special Products Section P.3 Polynomials and Special Products

Definition of a Polynomial in x Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree -2 5 1 3 14 0 Polynomial Function

Definition of a Polynomial in x Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form Where an ≠ 0. The polynomial is of degree n. The leading coefficient is an The constant term is a0.

Definition of a Polynomial in x Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form Where an ≠ 0. Is this a polynomial?

Definition of a Polynomial in x Let a0, a1, a2,…an be real numbers and let n be a nonnegative integer. A polynomial in x is an expression of the form Where an ≠ 0. Is this a polynomial?

Names of Polynomials x2 – 16 x + 8 x4 + x3 – x – 15 7 x2 – 6x + 5 2nd degree, binomial 1st degree, binomial 4th degree, polynomial Constant, monomial 2nd degree, trinomial 7th degree, binomial x2 – 16 x + 8 x4 + x3 – x – 15 7 x2 – 6x + 5 x4y3 + 7

FOIL F O I L

Square a Binomial Now here is a shortcut worth knowing.

Square a Binomial a2 Square the first term 2ab Double the product These are directions for a shortcut in binomial multiplication. a2 Square the first term 2ab Double the product of the terms b2 Square the last term

Square a Binomial Negative Version These are directions for a shortcut in binomial multiplication. a2 Square the first term. 2ab Double the product of the terms . Keep the sign b2 Square the last term Always positive.

Binomial Squared Examples A) (t + u)2 B) (2m - p)2 C) (4p + 3q)2 D) (5r - 6s)2 E) (3k - 1/2)2 t2 + 2tu + u2 4m2 - 4mp + p2 16p2 + 24pq + 9q2 25r2- 60rs + 36s2

The Product of the Sum and Difference of Two Terms First start with two terms. Now write two binomials. The other a difference One a sum

Binomial Sum and Difference Rule When multiplying two binomials that differ only in the sign between their terms, subtract the square of the last term from the square of the first term. Difference of Squares Conjugate Factors =

Sum and Difference Examples A) (6a + 3)(6a - 3) B) (10m + 7)(10m - 7) C) (7p + 2q)(7p - 2q) D) (3r - 1/2)(3r + 1/2) = 36a2 - 9 = 100m2 - 49 = 49p2 - 4q2

Cube a binomial

Apply the formula

Homework Page 30 1 – 6 9 – 42 multiples of 3 53, 59, 63, 71, 73, 86, 96, 97