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Section 6.5 Factoring Binomials

Review of factoring techniques: 1.ALWAYS look for a GCF first with any polynomial you are trying to factor. 2.After that, count how many terms are in the polynomial. If there are – FOUR terms: Use factoring by grouping. – THREE terms ( a trinomial): If the leading coefficient is 1 (if it looks like x 2 + bx + c), – Find factors (x + t)(x + v) such that t + v = b and t*v = c. If the leading coefficient is not 1 (if it looks like ax 2 + bx + c), – Use the British method, which starts with multiplying a*c, and then finding factors of that number that add up to b.

Case one: Binomials in which both terms are perfect squares. Previously, we discovered a formula for finding the product of the sum and difference of two terms: (a – b)(a + b) = a 2 – b 2 We can use the reverse of this equation to factor the difference of 2 squares. a 2 – b 2 = (a – b)(a + b) Today we are going to learn techniques for factoring two-term polynomials (binomials).

Examples: Factor x 2 – 16. Since this polynomial can be written as (x) 2 – (4) 2, x 2 – 16 = (x – 4)(x + 4). Factor 9x 2 – 4. Since this polynomial can be written as (3x) 2 – (2) 2, 9x 2 – 4 = (3x – 2)(3x + 2). Factor 16x 2 – 9y 2. Since this polynomial can be written as (4x) 2 – (3y) 2, 16x 2 – 9y 2 = (4x – 3y)(4x + 3y). a 2 – b 2 = (a – b)(a + b)

Examples Factor x 8 – y 6. Since this polynomial can be written as (x 4 ) 2 – (y 3 ) 2, x 8 – y 6 = (x 4 – y 3 )(x 4 + y 3 ). Factor x This one is the sum of two squares, not the difference of squares, so it can’t be factored. This polynomial is a prime polynomial. a 2 – b 2 = (a – b)(a + b)

Example: Remember that you should always factor out any common factors first, before you start any other technique. Step 1: Factor out the GCF, which in this case is 4. 36x 2 – 64 = 4(9x 2 – 16) Step 2: Factor the polynomial 9x 2 – 16 The polynomial can be written as (3x) 2 – (4) 2, so (9x 2 – 16) = (3x – 4)(3x + 4). Our final result is 36x 2 – 64 = 4(3x – 4)(3x + 4). ↑ (Don’t forget to write in 4 (the GCF) as part of your final answer!) Factor 36x 2 – 64. a 2 – b 2 = (a – b)(a + b)

Example from the online homework: Note that 4096 is a HUGE NUMBER. But if it is a square, then we’re in business. How do you tell if it’s a perfect square? Take the square root (You can do this with your on-line calculator.) Answer: 4096 = So s 4 – 4096 = (s 2 ) 2 – 64 2 = (s 2 – 64)(s ) Are we done yet? No, because s 2 – 64 factors further into (s + 8)(s – 8) Final answer: (s + 8)(s – 8)(s )

There are two additional types of binomials that can be factored easily by remembering a formula. We have not studied these special products previously, as they involve cubes of terms, rather than just squares. a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) [You can prove that these formulas work by multiplying out the two factors on the right side of each equation. If you do this, you’ll get six products at the first stage, but you’ll find that everything cancels out except the a 3 and b 3 terms.] NOTE: These formulas are on the formula sheet that you can use on each test and quiz, so you don’t have to memorize them, but you do need to know how to apply them.)

1.Factor x HINT: This one looks like ( ) 3 + ( ) 3. Since this polynomial can be written as (x) 3 + (1) 3, we can use the sum of cubes formula, with a = x and b = 1 Answer: x = (x + 1)(x 2 – x + 1). 2. Factor y 3 – 64. HINT: This one looks like ( ) 3 - ( ) 3. What goes in the parentheses? Therefore we can use the difference of cubes formula with a = y and b = 4. Answer: y 3 – 64 = (y – 4)(y 2 + 4y + 16). Examples Formulas: Sum of cubes: a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) Diff. of cubes: a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) HINT: It will help you a lot when doing cube problems if you always start by writing blank parentheses, then figure out what goes in them. Those will be the “a” and “b” in your formula. y4

3. Factor 8t 3 + s 6. HINT: This one looks like ( ) 3 + ( ) 3. What goes in the parentheses? This polynomial can be written as (2t) 3 + (s 2 ) 3, so 8t 3 + s 6 = (2t + s 2 )((2t) 2 – (2t)(s 2 ) + (s 2 ) 2 ) = (2t + s 2 )(4t 2 – 2s 2 t + s 4 ). 4. Factor x 3 y 6 – 27z 3. HINT: This one looks like ( ) 3 - ( ) 3. What goes in the parentheses? This polynomial can be written as (xy 2 ) 3 – (3z) 3, so x 3 y 6 – 27z 3 = (xy 2 – 3z)((xy 2 ) 2 + (3z)(xy 2 ) + (3z) 2 ) = (xy 2 – 3z)(x 2 y 4 + 3xy 2 z + 9z 2 ). Examples: a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) a 3 – b 3 = (a – b)(a 2 + ab + b 2 )

Remember to ALWAYS check to see if you can factor out any common factors before attempting to use any other factoring techniques or formulas. Step 1: Factor out the GCF. (Tip: Since 375 is such a big number, start by factoring the smaller number 24, then see if any of its factors will divide into 375. You will find that the number 3 is a divisor of both 24 and 375.) 375y 6 – 24y 3 = 3y 3 (125y 3 – 8) Since the second part can be written as (5y) 3 – 2 3, 125y 3 – 8 = (5y – 2)((5y) 2 + (5y)(2) ) = (5y – 2)(25y y + 4). Final answer: 3y 3 (5y – 2)(25y y + 4). Example Factor 375y 6 – 24y 3.

The assignment on this material (HW 6.5) Is due at the start of the next class session. You may now OPEN your LAPTOPS and begin working on the homework assignment.