Beginning Algebra 5.4 The Difference of Two Squares 5.5 The Sum and Difference of Two Cubes

6.4 Factoring Trinomials Objective 1. To factor the difference of squares. Objective 3. To factor a polynomial by first factoring out the greatest common factor and then factoring the polynomial that remains. Objective 2. To factor a perfect square trinomial.

difference of squares Some rules for factoring the difference of squares: 2. The powers must be even even whole numbers numbers. 1. The numbers (coefficients) must be squares squares. Factor Difference of Squares: 3. Binomials that are the sums of squares squares will be Prime Factors Factors.

(9x 2 – 49y 2 ), (a 2 b 2 – 9c 2 ), (4z 4 – 16), (x 8 – 81), etc. binomials Factorable binomials include: c) The difference difference of two even-powered terms (2 or higher) such as: (x 4 – y 4 ), (x 16 – y 16 ), (x 2 – y 6 ), (x 4 – y8)y8) Difference of Squares Factor Difference of Squares: a) The difference difference of a second degree term term with any real square number number, such as: (x 2 (x 2 – 4), (y 2 (y 2 – 9), (4z 2 (4z 2 – 25), (16 – x 2 ), x 2 ), etc. b) The difference difference of two even-powered terms (2 or higher) such as:

sum b) The sum of an even degree term (2 or higher) with any real number, such as: (x 2 + 4), (a 2 b 2 + 9b 2 ), (4z ), (x y 2 ), etc. binomial Prime binomial factors include: a) Linear Linear binomials binomials, such as: (x - 4), (a + b), (9 + y) c) The sum sum of two even-powered terms terms, such as: Difference of Squares Factor Difference of Squares: (x 4 + y 4 ), (x 16 + y 16 ), (x 2 + y 6 ), (x 4 + y 8 ), etc. d) Differences Differences that may contain squares squares, such as: (2x 2 – 3), (a 2 – 5), (a 2 – b 3 ), etc.

What to DoHow to Do It Product of special special binomials sum and difference: (4x + 7)(4x – 7) The sum and difference of the same two numbers numbers are called conjugate pairs Product of conjugate pairs: (4x + 7)(4x – 7) + 0 difference of squares. = 16x 2 – 49 (4x + 7)(4x – 7) 16x 2 F - 27x O 28x I – 49 L sum Note sum of O + I terms: Conjugates Review Multiplication of Conjugates

What to DoHow to Do It Product of special special binomials sum and difference: (2x + 3)(2x – 3) The sum and difference of the same two numbers numbers are called conjugate pairs Product of conjugate pairs: (2x + 3)(2x – 3) + 6x - 6x 4x – 9 difference of squares. = 4x 2 – 9 (2x + 3)(2x – 3) sum Note sum of O + I terms:

Conjugates Review Multiplication of Conjugates What to DoHow to Do It Product of special special binomials sum and difference: (3x + 5)(3x – 5) Product of conjugate pairs: (3x + 5)(3x – 5) + 15x - 15x 9x – 25 difference of squares. = 9x 2 – 25 (3x + 5)(3x – 5) sum Note sum of O + I terms: The sum and difference of the same two numbers numbers are called conjugate pairs

Conjugate Pairs  Pairs  Difference of Squares Binomial Conjugates Product of Binomial Conjugates conjugatesdifference squareof first termsquarelast term The product of conjugates is the difference of the square of the first term and square of the last term. (A + B)(A – B) = A 2 – B2B2 The sum and difference of the same two numbers are called conjugate pairs pairs.

Difference of Squares Conjugate Pairs differencesquares conjugates The factors of the difference of the squares are the product of conjugates. A 2 – B 2 = (A + B)(A – B) The sum sum and difference difference of the same two numbers are called conjugate pairs pairs. Difference of Squares Factor Difference of Squares:

Factor as sum sum and difference difference of bases of the squares squares. Given a binomial difference of squares squares: What to do How to do it A2 – B2A2 – B2 Difference of Squares Factor Difference of Squares: (A + B)(A – B) x2 – y2x2 – y2 Factor as sum sum and difference difference of bases of the squares squares. (x + y)(x – y) difference of squares Given a binomial difference of squares:

What to DoHow to Do It Given Second Degree Binomial Binomial: 4z 2 – 25 (2z) 2 – (5) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (2z + 5)(2x – 5) 4z 2 – 25 = (2z + 5)(2z – 5) Rewrite square factors factor(s) in each term. Factor as sum sum and difference of bases of the squares squares. The factors of difference difference of squares is the product of conjugate pairs pairs.

What to DoHow to Do It y 2 – 289 (y) 2 – (17) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (y + 17)(y – 17) y2 y2 – 289 = (y + 17)(y – 17) The factors of difference difference of squares is the product of conjugate pairs pairs. Given Second Degree Binomial Binomial: Rewrite square factors factor(s) in each term. Check if necessary. Factor as sum sum and difference of bases of the squares squares.

What to DoHow to Do It 9x 2 – 49y 2 (3x) 2 – (7y) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (3x + 7y)(2x – 7y) 9x 2 – 49y 2 = (3x + 7y)(2x – 7y) The factors of difference difference of squares is the product of conjugate pairs pairs. Given Second Degree Binomial Binomial: Rewrite square factors factor(s) in each term. Check if necessary. Factor as sum sum and difference of bases of the squares squares.

What to DoHow to Do It (ab) 2 – (3c) 2 difference of squares prime factors Difference of Squares Factor Difference of Squares: (ab + 3c)(ab – 3c) a2b2 a2b2 – 9c 2 = (ab + 3c)(ab – 3c) The factors of difference difference of squares is the product of conjugate pairs pairs. a 2 b 2 – 9c 2 Given Second Degree Binomial Binomial: Rewrite square factors factor(s) in each term. Check if necessary. Factor as sum sum and difference of bases of the squares squares.

What to DoHow to Do It x 8 – 81 (x 4 ) 2 – (9) 2 difference of squares Difference of Squares Factor Difference of Squares: (x 4 + 9)(x 2 + 3)(x 2 – 3) x8 x8 – 81 = (x 4 + 9)(x 2 + 3)(x 2 – 3) Factor second binomial binomial again. Complete factors of difference difference of squares squares require repeated factoring factoring. (x 4 + 9)(x 4 – 9) Given Second Degree Binomial Binomial: Rewrite square factors factor(s) in each term. Check [ Check if necessary. necessary. ] Factor as sum sum and difference of bases of the squares squares. difference of squares x4 – 9x4 – 9 relatively prime factors

What to DoHow to Do It x 4 – y 8 (x 2 ) 2 – (y 4 ) 2 difference of squares relatively prime factors Difference of Squares Factor Difference of Squares: (x 2 + y 4 )(x + y 2 )(x – y2)y2) x4 x4 – y 8 = (x 2 + y 4 )(x + y 2 )(x – y2)y2) Complete factors of difference difference of squares squares require repeated factoring factoring. (x 2 + y 4 )(x 2 – y4)y4) difference of squares x2 – y4x2 – y4 Factor second binomial binomial again. Given Second Degree Binomial Binomial: Rewrite square factors factor(s) in each term. Check if necessary. Factor as sum sum and difference of bases of the squares squares.

What to DoHow to Do It 3x 2 – 12 3(x 2 – 4) difference of squares prime factors Factor binomial as the difference of squares Multiply by common factor factor 3 left in composite or power form. Difference of Squares Factor Difference of Squares: (x + 2)(x – 2) Factor out the common factor(s) from each term. Given Second Degree Binomial Binomial: 3x 2 – 12 = 3(x + 2)(x – 2)

What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: a 2 x 2  2abx + b 2 = (ax  b) 2 (ax) 2  2abx + b2b2 twice ax twice b 2abx Perfect square trinomials must have the firstlast terms first and last terms be perfect squares last sign positive and the last sign positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign  is the sign of the binomial binomial. a 2 x 2  2abx + b 2 (ax  b) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms.

What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: 25x x + 36 = (5x + 6) 2 (5x) x Perfect square trinomials Perfect square trinomials must have the firstlast terms first and last terms be perfect squares last sign positive and the last sign positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign + is the sign of the binomial binomial. 25x x + 36 (5x + 6) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms. twice 5x twice 6 2  5x  6 60x 60x

What to DoHow to Do It Given Perfect Square Trinomial Trinomial: Perfect Square Trinomial Factor Perfect Square Trinomial: 16x x + 49 = (4x - 7) 2 (4x) 2 – 56x Perfect square trinomials Perfect square trinomials must have the firstlast terms first and last terms be perfect squares last sign positive and the last sign positive. If all of these conditions hold, check to see if the middle term is twice twice the product of square roots roots of first term term and last term term. The middle sign sign – is the sign of the binomial binomial. 16x 2 – 60x + 49 (4x – 7) 2 Factors square rootsfirst last terms Factors of the square of the binomial with square roots of first and last terms. twice 4x twice 7 2  4x  7 60x 60x

Sum or Difference of Cubes a 3 – b 3 = (a – b) (a 2 + ab + b 2 ) a 3 + b 3 = (a + b) (a 2 - ab + b 2 )

Factoring Sums & Differences of Cubes x =

x = (x) 3 + (4) 3

x = x = (x) 3 + (4) 3 = (x + 4) (x 2 - 4x+ 16)

Factoring Sums & Differences of Cubes z 3 - 1

z = (z) 3 – (1) 3

z z = (z) 3 – (1) 3 = (z - 1) (z 2 + z + 1)

27x Factor

27x Factor

27x Factor

THE END 5.4