Polynomials & Factoring Chapter 10

Chapter 10 Chapter Goals Identify, compute with, & factor polynomial expressions. Add, subtract, & multiply polynomials Factoring polynomials Solve polynomials equations by factoring

Chapter Study Guide Chapter 10 10.1 – Adding & Subtracting Polynomials 10.2 – Multiply Polynomials 10.3 – Special Products of Polynomials 10.4 – Solving Quadratic Equations in Factored Form 10.5 – Factoring x2 + bx + c 10.6 – Factoring ax2 + bx + c 10.7 – Factoring Special Products 10.8 – Factoring Cubic Polynomials

Key Words – Define these Chapter 10 Key Words – Define these Monomial FOIL pattern Degree of a monomial Factored form Polynomial Zero-product property Binomial Factor a trinomial Trinomial Perfect square trinomial Standard form Prime polynomial Degree of a polynomial Factor completely

Adding & Subtracting Polynomial 10.1 Adding & Subtracting Polynomial A monomial is a number, a variable, or the product of a number & one or more variables with whole number exponents. The following expressions are monomials: 8, -2x, 3x2y The degree of a monomial is the sum of the exponents of the variables in the monomial. The degree of 3x2 is 2. The degree of - 6z4 is 4. The degree of 3x2y is 2 + 1, or 3.

Adding & Subtracting Polynomial: Cont. 10.1 Adding & Subtracting Polynomial: Cont. Polynomials – a polynomial is a monomial or a sum of monomials. A polynomial such as x2 + (-4x) + (-5) is usually written as x2 – 4x – 5. Each of the following expressions is a polynomial. A polynomial of two terms is a binomial. A polynomial of three terms is a trinomial. Polynomials are usually written in standard form, which means that the terms are arranged in decreasing order, from largest exponent to smallest exponent. The degree of a polynomial in one variable is the largest exponent of that variable.

Multiplying Polynomials 10.2 Multiplying Polynomials (x + 4)(x +5) First distribute the binomial (x + 5) to each term of (x + 4). (x + 4)(x +5) = x(x + 5) + 4(x + 5) Then distribute the x & the 4 to each term of (x + 5) = x(x) + x(5) + 4(x) + 4(5) = x2 + 5x + 4x + 20 = x2 + 9x + 20

Multiplying Polynomials: Cont. 10.2 Multiplying Polynomials: Cont. FOIL Pattern – In using the distributive property for multiplying two binomials, you may have noticed the following pattern. Multiply the First, Outer, Inner, & Last terms. Then combine like terms. This pattern is called the FOIL pattern. To multiply two polynomials that have three or more terms, remember that each term of one polynomial must be multiplied by each of the other polynomial. Use a vertical or a horizontal format. Write each polynomial in standard form.

Special Products of Polynomials 10.3 Special Products of Polynomials Some pairs of binomials have special products. If you learn to recognize such pairs, finding the product of two binomials will sometimes be quicker & easier. For example, to find the product of (y – 3) (y + 3), you could multiply the two binomials using the FOIL pattern (y + 3)(y – 3) = y2 + (-3y) + 3y – 9 = y2 – 9 Sum & Difference Pattern (a + b)(a – b) = a2 – b2

Special Products of Polynomials: Cont. 10.3 Special Products of Polynomials: Cont. To find the product of (x + 4)2 (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 Square of a Binomial Pattern (a + b)2 = a2 + 2ab + b2 or (a – b)2 = a2 – 2ab + b2

Special Products of Polynomials: Cont. 10.3 Special Products of Polynomials: Cont. Area Models – Area models may be helpful when multiplying two binomials or using any of the special patterns. The square of a binomial pattern (a + b)2 = a2 + 2ab + b2 can be modeled as shown below The area of a large square is (a +b)2, which is equal to the sum of the areas of the two small squares and two rectangles. Notice that the two rectangles with area ab produce the middle term 2ab

Solving Quadratic Equations in Factored Form 10.4 Solving Quadratic Equations in Factored Form A polynomial is in factored form if it is written as the product of two or more factors. The polynomials in the following equations are written in factored form. x(x – 7) = 0 (x + 2)(x + 5) = 0 (x + 1)(x – 3)(x + 8) = 0 A value of x that makes any of the factors zero is a solution of the polynomial equation That these are the only solutions follows from the zero-product property, stated below. Zero-product property Let a & b be real numbers. If ab = 0, then a = 0 or b = 0 If the product of two factors is zero, then at least one of the factors must be zero.

10.5 Factoring x2 + bx + c To factor a trinomial of this form means to write the trinomial as the product of two binomials (factored form) In order to write x2 + bx + c in the form (x + p)(x + q), note that (x + p)(x + q) = x2 + (p + q)x + pq This leads you to seek numbers p & q such that p + q = b & pq = c

10.6 Factoring ax2 + bx + c One way to factor a trinomial of the form ax2 + bx + c is to find numbers m & n whose product is a & numbers p & q whose product is c so that the middle term is the sum of the Outer & Inner products of FOIL. ax2 + bx + c = (mx + p)(nx +q) m x n = a p x q = c b = mq + np

Factoring Special Products 10.7 Factoring Special Products Difference of Two Square Patterns a2 – b2 = (a + b)(a – b) Perfect Square Trinomial Pattern a2 + 2ab + b2 = (a + b)2 a2 – 2ab +b2 = (a – b)2

Factoring Cubic Polynomials 10.8 Factoring Cubic Polynomials Factor out the common factor 9x2 – 15 = 3(3x2 – 5) You can also factor out variable factors that are common to the terms of a polynomial. When factoring a cubic polynomial, you should factor out the greatest common factor (GCF) first & then look for other patterns.

Factoring Cubic Polynomials 10.8 Factoring Cubic Polynomials Prime Factors – A polynomial is prime if it cannot be factored using integer coefficients. To factor a polynomial completely, write it as the product of monomial & prime factors. Factoring by Grouping – Another use of the distributive property is in factoring polynomials that have four terms. Sometimes you can factor the polynomial by grouping the terms into two groups & factoring the greatest common factor out of each term. Sum of Two Cubes Pattern a3 + b3 = (a + b)(a2 – ab + b2) Difference of Two Cubes Pattern a3 – b3 = (a – b)(a2 + ab + b2)

Factoring Cubic Polynomials: Cont. 10.8 Factoring Cubic Polynomials: Cont. Pattern Used to Solve Polynomial Equations Graphing: Can be used to solve any equation, but gives only approximate solutions. The Quadratic Formula: Can be used to solve any quadratic equation Factoring: Can be used with the zero-product property to solve an equation that is in standard form & whose polynomial is factorable a2 – b2 = (a + b)(a – b) a2 + 2ab + b2 = (a + b)2 a2 – 2ab + b2 = (a – b)2 a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2)