3After this presentation you should be able to: Explain the concept of factoring polynomialsExplain in simple terms why you would need to factor a polynomialFactor polynomials that have a greatest common factor between terms. (ex. x3 + 3x2 +x)Factor polynomials that have a leading coefficient of 1(ex. x2 + 2x +1)Factor polynomials that have a leading coefficient other then 1 (ex. 3x2 + 13x +4)
4Factoring a polynomial is the same in concept to factoring a monomial For the monomial 6x2y, the prime factors are ·3·x·x·yFor the polynomial x3 + 2x2 + x, the prime factors are x and (x2 + x + 1) written as x(x2 + x + 1)This presentation will show you how to factor some basic polynomialsFactoring a polynomial is the same in concept to factoring a monomial. For example, the factors of the monomial 6x2y are the prime components of the term written as a product. That is 6x2y is broken down (or factored) into 2·3· x·x·y
5Why factor polynomials? 1) To reduce expressions to their lowest terms like you do with monomials:Monomial Example:6x2y = 2·3· x·x·y = 3xy2x ·xPolynomial Example:x3 + 2x2 + x = x·(x2 + x + 1) = xx2 + x (x2 + x + 1)2) Factoring polynomials is key to solving quadratics and other higher order equations.
6SummaryFactoring a polynomial means expressing it as a product of other polynomials. This can help you simplify expressions and solve higher order equations
7Factoring polynomials with a Greatest Common Factor (GCF) Factoring Method #1Factoring polynomials with a Greatest Common Factor (GCF)IMPORTANT - Always look for a GCF before using any other factoring method.
8Steps: Find the greatest common factor (GCF). Divide the polynomial by the GCF. The quotient (answer from the division) is the other factor.Express the polynomial as the product of the quotient and GCF.
12A note: Method #2 is a way of ‘working backwards’ from the way one would multiply two binomials togetherNotice:The first term of the answer is the product of the (first) variable terms.The last term of the answer is the product of the constant (last) terms.The middle term is the sum of the constant (last) termsPlease keep this in mind as we go forward…
13Factoring Polynomials – Method #2 How do you factor a trinomial with a leading coefficient of 1?Example: Factor x2 – 13x + 36You can use a diamond...Write the middle coefficient here–13Now, find factors that will multiply to the bottom number, and add to the top number.– –4+36Write the last term here.The factors are (x – 9)(x – 4)Example: Factor x2 – 3x – 40–3–40The factors are (x – 8)(x + 5)–
14Try factoring these on your own: (x - 3)(x + 5)1) x2 + 2x – 15 =(x - 4)(x - 6)2) x =3) x2 + 6x + 9 =(x + 3)(x + 3) = (x + 3)2
15Some polynomials are already prime there are no factors Look at x2 + 5x + 1Are there two numbers that multiply to get 1 and add to get 5?No - There are not two integers that will do this and so this quadratic doesn’t factor. Remember this as a possibility when factoring any polynomial
16How do you factor a trinomial whose leading coefficient is not 1? Factoring Polynomials – Method #3How do you factor a trinomial whose leading coefficient is not 1?Example: Factor 3x2 + 13x + 4We will make a T to determine the coefficients of the factors...Write factors ofthe first term inthis column.Write factors ofthe last term inthis column.( )3123114These are thecoefficients ofthe factors...( )≠ 13= 13Multiple diagonally and add.See if the sum matches themiddle term...…if not, try anothercombination of factors...(3x + 1)(x + 4)
17Example: Factor 6d2 + 33d – 63Remember, look for the GCF first...GCF: 33(2d2 + 11d – 21)Now, factor the trinomial (using a T)2–317–3 + 14= 113(2d – 3)(d + 7)
19Closing NotesRemember – whenever you factor any polynomial, your first step is to see if there is a greatest common factor.After completing an initial factorization, check to see if any of the factors can be factored further (see example of slide #13).Don’t get stuck on a polynomial that cannot be factored – there are prime polynomials just as there are prime numbers that cannot be factored. Eliminate all possibilities then move on.