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Sample Presentation on Factoring Polynomials By Paul Bartholomew K12 International Academy Applicant (856)

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After this presentation you should be able to: Explain the concept of factoring polynomials Explain in simple terms why you would need to factor a polynomial Factor polynomials that have a greatest common factor between terms. (ex. x 3 + 3x 2 +x) Factor polynomials that have a leading coefficient of 1 (ex. x 2 + 2x +1) Factor polynomials that have a leading coefficient other then 1 (ex. 3x x +4)

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Factoring a polynomial is the same in concept to factoring a monomial For the monomial 6x 2 y, the prime factors are 2·3·x·x·y For the polynomial x 3 + 2x 2 + x, the prime factors are x and (x 2 + x + 1) written as x(x 2 + x + 1) This presentation will show you how to factor some basic polynomials

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Why factor polynomials? 1) To reduce expressions to their lowest terms like you do with monomials: 6x 2 y = 2·3· x·x·y = 3xy 2x 2·x Monomial Example: Polynomial Example: x 3 + 2x 2 + x = x·(x 2 + x + 1) = x x 2 + x + 1 (x 2 + x + 1) 2) Factoring polynomials is key to solving quadratics and other higher order equations.

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Factoring a polynomial means expressing it as a product of other polynomials. This can help you simplify expressions and solve higher order equations Summary

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Factoring Method #1 Factoring polynomials with a Greatest Common Factor (GCF) IMPORTANT - Always look for a GCF before using any other factoring method.

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Steps: 1)Find the greatest common factor (GCF). 2)Divide the polynomial by the GCF. The quotient (answer from the division) is the other factor. 3)Express the polynomial as the product of the quotient and GCF.

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The answer should look like this:

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STEP #1: Step#2: Divide.

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Factor these on your own looking for a GCF.

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A note: Method #2 is a way of ‘working backwards’ from the way one would multiply two binomials together Notice: 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) terms Please keep this in mind as we go forward…

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Factoring Polynomials – Method #2 How do you factor a trinomial with a leading coefficient of 1? Example: Factor x 2 – 13x + 36 You can use a diamond... The factors are (x – 9)(x – 4) Write the middle coefficient here Write the last term here. – Now, find factors that will multiply to the bottom number, and add to the top number. –9 –4 Example: Factor x 2 – 3x – 40 –3 –40 –8 +5 The factors are (x – 8)(x + 5)

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Try factoring these on your own: 1) x 2 + 2x – 15 = 2) x = 3) x 2 + 6x + 9 = (x - 3)(x + 5) (x - 4)(x - 6) (x + 3)(x + 3) = (x + 3) 2

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Some polynomials are already prime there are no factors Look at x 2 + 5x + 1 Are 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

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How do you factor a trinomial whose leading coefficient is not 1? (3x + 1)(x + 4) Example: Factor 3x x + 4 We will make a T to determine the coefficients of the factors... Write factors of the first term in this column Write factors of the last term in this column Multiple diagonally and add. See if the sum matches the middle term ≠ 13 …if not, try another combination of factors = 13 ( ) These are the coefficients of the factors... Factoring Polynomials – Method #3

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Example: Factor 6d d – 63 Remember, look for the GCF first... GCF: 3 3(2d d – 21) Now, factor the trinomial (using a T) 3(2d – 3)(d + 7) – –3 7 = 11

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Try factoring these on your own: 1) 3x 2 - 2x – 8 = 2) 5x 2 – 17x + 6 = 3) 4x x – 6 = (3x - 4)(x + 2) (x - 3)(x + 5)

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Closing Notes Remember – 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.

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