10.1 Adding and Subtracting Polynomials A polynomial of two terms is a binomial. 7xy2 + 2y A polynomial of three terms is a trinomial. 8x2 + 12xy + 2y2 The leading coefficient of a polynomial is the coefficient of the variable with the largest exponent. The constant term is the term without a variable. The degree is 3. 6x3 – 2x2 + 8x + 15 The leading coefficient is 6. The constant term is 15.

10.1 Adding and Subtracting Polynomials The degree of a polynomial is the greatest of the degrees of any of its terms. The degree of a term is the sum of the exponents of the variables. Examples: 3y2 + 5x + 7 degree 2 21x5y + 3x3 + 2y2 degree 6 Common polynomial functions are named according to their degree. Degree Function Equation linear f (x) = mx + b one f (x) = ax2 + bx + c, a  0 quadratic two cubic three f (x) = ax3 + bx2 + cx + d, a  0

10.1 Adding and Subtracting Polynomials To add polynomials, combine like terms. Examples: Add (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10). Use a horizontal format. Rearrange and group like terms. (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10) = (5x3 + 3x3 ) + (6x2 – 12x2) + (3 – 10) = 8x3 – 6x2 – 7 Combine like terms.

10.1 Adding and Subtracting Polynomials Add (6x3 + 11x –21) + (2x3 + 10 – 3x) + (5x3 + x – 7x2 + 5). Use a vertical format. 6x3 + 11x – 21 Arrange terms of each polynomial in descending order with like terms in the same column. 2x3 – 3x + 10 5x3 – 7x2 + x + 5 13x3 – 7x2 + 9x – 6 Add the terms of each column.

10.1 Adding and Subtracting Polynomials The additive inverse of the polynomial x2 + 3x + 2 is – (x2 + 3x + 2). This is equivalent to the additive inverse of each of the terms. – (x2 + 3x + 2) = – x2 – 3x – 2 To subtract two polynomials, add the additive inverse of the second polynomial to the first.

10.1 Adding and Subtracting Polynomials Example: Add (4x2 – 5xy + 2y2) – (– x2 + 2xy – y2). (4x2 – 5xy + 2y2) – (– x2 + 2xy – y2) Rewrite the subtraction as the addition of the additive inverse. = (4x2 – 5xy + 2y2) + (x2 – 2xy + y2) = (4x2 + x2) + (– 5xy – 2xy) + (2y2 + y2) Rearrange and group like terms. = 5x2 – 7xy + 3y2 Combine like terms.

10.1 Adding and Subtracting Polynomials Let P(x) = 2x2 – 3x + 1 and R(x) = – x3 + x + 5. Examples: Find P(x) + R(x). P(x) + R(x) = (2x2 – 3x + 1) + (– x3 + x + 5) = – x3 + 2x2 + (– 3x + x) + (1 + 5) = – x3 + 2x2 – 2x + 6

10.2 Multiplying Polynomials To multiply a polynomial by a monomial, use the distributive property and the rule for multiplying exponential expressions. Examples:. Multiply: 2x(3x2 + 2x – 1). = 2x(3x2 ) + 2x(2x) + 2x(–1) = 6x3 + 4x2 – 2x

10.2 Multiplying Polynomials Multiply: – 3x2y(5x2 – 2xy + 7y2). = – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2) = – 15x4y + 6x3y2 – 21x2y3

10.2 Multiplying Polynomials To multiply two polynomials, apply the distributive property. Example: Multiply: (x – 1)(2x2 + 7x + 3). = (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3) = 2x3 – 2x2 + 7x2 – 7x + 3x – 3 = 2x3 + 5x2 – 4x – 3

10.2 Multiplying Polynomials Example: Multiply: (x – 1)(2x2 + 7x + 3). Two polynomials can also be multiplied using a vertical format. 2x2 + 7x + 3 x – 1 Example: – 2x2 – 7x – 3 Multiply – 1(2x2 + 7x + 3). 2x3 + 7x2 + 3x Multiply x(2x2 + 7x + 3). 2x3 + 5x2 – 4x – 3 Add the terms in each column.

10.2 Multiplying Polynomials To multiply two binomials use a method called FOIL, which is based on the distributive property. The letters of FOIL stand for First, Outer, Inner, and Last. 1. Multiply the first terms. 4. Multiply the last terms. 2. Multiply the outer terms. 5. Add the products. 3. Multiply the inner terms. 6. Combine like terms.

10.2 Multiplying Polynomials Examples: Multiply: (2x + 1)(7x – 5). First Outer Inner Last = 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5) = 14x2 – 10x + 7x – 5 = 14x2 – 3x – 5

10.2 Multiplying Polynomials Multiply: (5x – 3y)(7x + 6y). First Outer Inner Last = 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y) = 35x2 + 30xy – 21yx – 18y2 = 35x2 + 9xy – 18y2

10.3 Special Cases The multiply the sum and difference of two terms, use this pattern: (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2 square of the second term square of the first term

10.3 Special Cases Examples: (3x + 2)(3x – 2) (x + 1)(x – 1)

10.3 Special Cases To square a binomial, use this pattern: (a + b)2 = (a + b)(a + b) = a2 + ab + ab + b2 = a2 + 2ab + b2 square of the first term twice the product of the two terms square of the last term

10.3 Special Cases Examples: Multiply: (2x – 2)2 . = (2x)2 + 2(2x)(– 2) + (– 2)2 = 4x2 – 8x + 4 Multiply: (x + 3y)2 . = (x)2 + 2(x)(3y) + (3y)2 = x2 + 6xy + 9y2

10.4 Factoring The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term. Example: Factor 18x3 + 60x. GCF = 6x Find the GCF. 18x3 + 60x = 6x (3x2) + 6x (10) Apply the distributive law to factor the polynomial. = 6x (3x2 + 10) Check the answer by multiplication. 6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x

10.4 Factoring Example: Factor 4x2 – 12x + 20. Therefore, GCF = 4. 4x2 – 12x + 20 = 4x2 – 4 · 3x + 4 · 5 = 4(x2 – 3x + 5) Check the answer. 4(x2 – 3x + 5) = 4x2 – 12x + 20

10.4 Factoring A common binomial factor can be factored out of certain expressions. Example: Factor the expression 5(x + 1) – y(x + 1). 5(x + 1) – y(x + 1) = (5 – y)(x + 1) Check. (5 – y)(x + 1) = 5(x + 1) – y(x + 1)

10.4 Factoring A difference of squares can be factored using the formula a2 – b2 = (a + b)(a – b). Example: Factor x2 – 9y2. x2 – 9y2 = (x)2 – (3y)2 Write terms as perfect squares. = (x + 3y)(x – 3y) Use the formula.

10.4 Factoring The same method can be used to factor any expression which can be written as a difference of squares. Example: Factor 4(x + 1)2 – 25y 4. 4(x + 1)2 – 25y 4 = (2(x + 1))2 – (5y2)2 = [(2(x + 1)) + (5y2)][(2(x + 1)) – (5y2)] = (2x + 2 + 5y2)(2x + 2 – 5y2)

10.4 Factoring Some polynomials can be factored by grouping terms to produce a common binomial factor. Examples: Factor 2xy + 3y – 4x – 6. 2xy + 3y – 4x – 6 = (2xy + 3y) – (4x + 6) Group terms. = (2x + 3)y – (2x + 3)2 Factor each pair of terms. = (2x + 3)( y – 2) Factor out the common binomial.

10.4 Factoring Factor 2a2 + 3bc – 2ab – 3ac. 2a2 + 3bc – 2ab – 3ac = 2a2 – 2ab + 3bc – 3ac Rearrange terms. = (2a2 – 2ab) + (3bc – 3ac) Group terms. = 2a(a – b) + 3c(b – a) Factor. = 2a(a – b) – 3c(a – b) b – a = – (a – b). = (2a – 3c)(a – b) Factor.

10.4 Factoring To factor a trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example, x2 + 10x + 24 = (x + 4)(x + 6).

10.4 Factoring One method of factoring trinomials is based on reversing the FOIL process. Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b. Example: Factor x2 + 3x + 2. x2 + 3x + 2 = (x + a)(x + b) F O I L = x2 + ax + bx + ab Apply FOIL to multiply the binomials. = x2 + (a + b) x + ab Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2. = x2 + (1 + 2) x + 1 · 2 Therefore, x2 + 3x + 2 = (x + 1)(x + 2).

10.4 Factoring = (x + a)(x + b) Example: Factor x2 – 8x + 15. = x2 + (a + b)x + ab Therefore a + b = – 8 and ab = 15. It follows that both a and b are negative. x2 – 8x + 15 = (x – 3)(x – 5). Check: (x – 3)(x – 5) = x2 – 5x – 3x + 15 = x2 – 8x + 15.

10.4 Factoring Example: Factor x2 + 13x + 36. = (x + a)(x + b) = x2 + (a + b) x + ab Therefore a and b are two positive factors of 36 whose sum is 13. = (x + 4)(x + 9) x2 + 13x + 36 Check: (x + 4)(x + 9) = x2 + 9x + 4x + 36 = x2 + 13x + 36.

10.4 Factoring Example: Factor 4x3 – 40x2 + 100x. 4x3 – 40x2 + 100x The GCF is 4x. = 4x(x2 – 10x + 25) Use distributive property to factor out the GCF. = 4x(x – 5)(x – 5) Factor the trinomial. Check: 4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25) = 4x(x2 – 10x + 25) = 4x3 – 40x2 + 100x

10.4 Factoring Example: Factor 2x2 + 5x + 3. 2x2 + 5x + 3 = (2x + a)(x + b) For some a and b. = 2x2 + (a + 2b)x + ab 2x2 + 5x + 3 = (2x + 3)(x + 1) Check: (2x + 3)(x + 1) = 2x2 + 2x + 3x + 3 = 2x2 + 5x + 3.

10.4 Factoring Example: Factor 4x2 – 12x + 5. Since ab = + 5, a and b have the same sign. This polynomial factors as (x + a)(4x + b) or (2x + a)(2x + b). a = –1, b = – 5 or a = 1 and b = 5. The middle term –12x equals either (4a + b) x or (2a + 2b) x. Since a and b cannot both be positive, they must both be negative. 4x2 – 12x + 5 = (2x –1)(2x – 5)

Many trinomials cannot be factored. Trinomials which are quadratic in form are factored like quadratic trinomials. Example: Factor 3x 4 + 28x2 + 9. 3x 4 + 28x2 + 9 = 3u2 + 28u + 9 Let u = x2. = (3u + 1)(u + 9) Factor. = (3x2 + 1)(x2 + 9) Replace u by x2. Many trinomials cannot be factored. Example: Factor x2 + 3x + 5. Let x2 + 3x + 5 = (x + a)(x + b) = x2 + (a + b) x + ab. Then a + b = 3 and ab = 5. This is impossible. The trinomial x2 + 3x + 5 cannot be factored.

Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor the first two terms: 6mx – 4m = 2m (3x - 2) Factor the last two terms: + 3rx – 2r = r (3x - 2) The green parentheses are the same so it’s the common factor Now you have a common factor (3x - 2) (2m + r)

Example: The length of a rectangle is (x + 5) ft Example: The length of a rectangle is (x + 5) ft. The width is (x – 6) ft. Find the area of the rectangle in terms of the variable x. x – 6 x + 5 A = L · W = Area L = (x + 5) ft W = (x – 6) ft A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30 = x2 – x – 30 The area is (x2 – x – 30) ft2.