1.3 Polynomials and Factoring

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1 1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable. Degree of Term: the sum of the exponents on the variables of the term. Monomial: a single term. Binomial: the sum or difference of two terms. Trinomial: the sum or difference of three terms. Polynomial: the sum or difference of many terms. Degree of Polynomial: The largest of the degrees of the individual terms. Polynomials of a Single Variable A polynomial in the variable x of degree n can be written in the form a n x n + a n-1 x n a 1 x + a 0 where a n, a n-1,, a 1, a 0 are number, an 0, and n is a positive integer. This form is called descending order, because the powers descend from left to right. The leading coefficient of this polynomial is a n. You can add two polynomials by combining like terms. Example 1 Add or subtract the polynomials as indicated. (4x 3 9x 2 + 1) + (-2x 3 8) (5x 2 6x + 2) (4 6x 3x 2 )

2 Multiplying Two Polynomials To multiply two polynomials, multiply each term of the first polynomial by each term of the second polynomial, using the distributive property. To multiply two binomials, use FOIL. F: First O: Outside I: Inside L: Last Example 2 Multiply the polynomials. (x 2 2y)(x 2 + y) (x + xy + y)(x y) Special Product Formulas (A B)(A + B) = A 2 B 2 (A + B) 2 = A 2 + 2AB + B 2 (A B) 2 = A 2 2AB + B 2 (A + B) 3 = A 3 + 3A 2 B + 3AB 2 + B 3 (A B) 3 = A 3 3A 2 B + 3AB 2 + B 3

3 Common Factoring Methods Factoring: reversing the process of multiplication in order to find two or more expressions whose product is the original expression. Factorable: a polynomial with integer coefficients that can be written as a product of two or more polynomials, all of which have also have integer coefficients. Prime/Irreducible: a polynomial that cannot be factored. Completely factor: to write a polynomial as a product of prime polynomials. Greatest Common Factor Method Factoring out those factors common to all the terms in an expression. This is the least complex of the factoring methods to apply. The Greatest Common Factor (GCF) among all the terms is the product of all the factors common to each term. Example 3 Use the Greatest Common Factor method to factor the following polynomials. (x 3 y) 2 (x 3 y) 27x 7 y + 9x 6 y 9x 4 yz

4 Factoring by Grouping Method Factoring a Four-Term Polynomial Arrange the terms so that the first two terms have a common factor and the last two terms have a common factor. For each pair of terms, use the distributive property to factor ou the pairs GCF. If there is now a common binomial factor, factor it out. If there is no common binomial factor in Step 3, begin again, rearranging the terms differently. Example 4 Factor each polynomial by grouping. ax 2bx 2ay + 4by x 2 + 3xy + 3y + x Factoring Special Binomials Method In the following, A and B represent algebraic expressions. Difference of Two Squares: A 2 B 2 = (A B)(A + B) Difference of Two Cubes: A 3 B 3 = (A B)(A 2 + AB + B 2 ) Sum of Two Cubes: A 3 + B 3 = (A + B)(A 2 AB + B 2 ) Note: A 2 + B 2 cannot be factored in the real sense.

5 Example 5 Factor the following special binomials. 27a 9 + 8b 12 25x 2 16y 2 Factoring Trinomials: Leading Coefficient is 1 Trinomial is in the form x 2 + bx + c. We must find two numbers, p and q, such that p + q = b and p q = c. The factored form of x 2 + bx + c would then be (x + p)(x + q) Example 6 Factor the following trinomials. x 2 + 6x + 9 x 2 5x + 6 x 2 + 5x + 4

6 Factoring Trinomials: Leading Coefficient is 1 For the trinomial ax 2 + bx + c, Multiply a and c. Factor ac into two integers whose sum is b (m + n = b, m n = a c). If no such factors exist, then the trinomial is irreducible over the integers (prime). Rewrite b in the trinomial with the sum found in step 2, and distribute. The resulting polynomial of four terms may now be factored by grouping. ax 2 + bx + c = ax 2 + (m + n)x + c = ax 2 + mx + nx + c = (ax 2 + mx) + (nx + c) Example 7 Factor the following Trinomials. 5a 2 37a 24 5x x 18 Factoring Expressions Containing Fractional Exponents Identify the least exponent among the various terms. Factor the variable raised to the least exponent from each of the terms. Factor out any other common factors. Simplify if possible.

7 Example 8 Factor the following algebraic expressions. 2x x -1 (3z + 2) 5/3 (3z + 2) 2/3 1.3 Homework # 1 57 every other odd