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Beginning Algebra

Beginning Algebra. 5.3 More Trinomials to Factor. 6.3 More Trinomials to Factor. Objective 1. To factor a trinomial whose leading coefficient is other than 1.

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Beginning Algebra

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  1. BeginningAlgebra 5.3 More Trinomials to Factor

  2. 6.3 More Trinomials to Factor Objective 1. To factor a trinomial whose leading coefficient is other than 1. Objective 2. To factor a polynomial by first factoring out the greatest common factor and then factoring the polynomial that remains.

  3. Same sign + Sum Larger sign – Difference Clue of Signs Ax2 Bx  C Read  as +or–

  4. Grouping NumberGN = P Same sign + Sum Larger sign - Difference Ax2 Bx  C A  C = P P =r  s, r > s r + s = B Ax2 Bx  C or r - s = B

  5. What to do How to do it or r - s = B Factor general quadratic trinomial: Ax2 Bx  C Given a general quadratic trinomial: 1. Find the product of the first and last coefficients: A and C AC= P 2. Find all of the pairs of factorsr and s P = rs, r > s r + s = B 3. so their sum or difference is the middle coefficient . is the middle coefficient .

  6. What to Do How to Do It Sum Same sign +   Larger sign Difference Ax2 rx  sx + C difference Factor by Clue of Signs: Given general trinomial of type that has nocommon factor. Read the clues of the signs. Ax2  Bx  C [Read  as “+ or” ] The productP = AC is the grouping number P = rs , r > s Find all possible factors of GN = P + sum (r + s) = B whose sum or difference is B (r ‑s) = B Rewrite middle termBx: and factor by grouping (ax  b)(cx  d)

  7. What to do How to do it 15 - 4 =11 Example: 10x2 + 11x - 6 1. Find the product of 10 and 6: 10 · 6 = 60 2. Find all of the pairs of factors: r and s 60 = 60 · 1 30 · 2 20 · 3 with the difference = 11. 15 · 4 12 · 5 10 · 6 3. Middle sign is + therefore: +15 , - 4 4. Separate middle term11x: +15x , - 4x

  8. What to do How to do it 10x2+ 15x - 4x - 6 Example: 10x2 + 11x - 6 10x2 + 11x - 6 5. Copy the polynomial: 6. Rewrite middle term11x: and group for factoring 7. Factor each group:bring down middle sign 5x(2x + 3) -2(2x + 3) (5x - 2)(2x + 3) 8. Factor common factor:

  9. What to Do How to Do It - 4x +15x 10x2 - 6 10x2+ 11x  6 = (5x  2)(2x + 3) Check Factors using FOIL Check by multiplying back using F0IL First (5x - 2)(2x + 3) Outer Inner Last 10x2+ 15x-4x- 6 Note sum of O + I terms

  10. What to do How to do it 9 - 2 = 7 Example: 3x2 - 7x - 6 1. Find the product of 3 and 6: 3 · 6 = 18 2. Find all of the pairs of factors: r and s 18 = 18 · 1 9 · 2 with the difference = 7. 6 · 3 3. Middle sign- is larger sign: - 9 , + 2 - 9x , + 2x 4. Separate middle term- 7x:

  11. What to do How to do it 3x2- 9x+ 2x - 6 Example: 3x2 - 7x - 6 3x2 - 7x - 6 5. Copy the polynomial: 6. Rewritemiddle term -7x: and group for factoring 7. Factor each group:bring down middle sign 3x(x - 3) +2(x - 3) (3x + 2)(x - 3) 8. Factor common factor:

  12. What to Do How to Do It +2x - 9x 3x2 - 6 3x2- 11x - 6 = (3x + 2)(x - 3) Check Factors using FOIL Check by multiplying back using F0IL First (3x + 2)(x - 3) Outer Inner Last 3x2- 9x+2x - 6 Note sum of O + I terms

  13. What to do How to do it 9 + 2 = 11 Example: 3x2 - 11xy + 6y2 1. Look at numbers only Find the product of 3 and 6: 3 · 6 = 18 18 = 18 · 1 2. Find all of the pairs of factors: r and s 9 · 2 with the sum = 11. 6 · 3 3. Middle sign- is same sign: - 9 , - 2 - 9xy , - 2xy 4. Separate middle term- 11xy:

  14. What to do How to do it 3x2- 9xy- 2xy + 6y2 Example: 3x2 - 11xy + 6y2 3x2 - 11xy + 6y2 5. Copy the polynomial: 6. Rewrite middle term-11xy: and group for factoring 7. Factor each group:bring down middle sign 3x(x - 3y) -2(x - 3y) (3x - 2y)(x - 3y) 8. Factor common factor:

  15. What to Do How to Do It - 2xy - 9xy 3x2 +6y2 3x2 - 11xy + 6y2 = (3x - 2)(x - 3) Check Factors using FOIL Check by multiplying back using F0IL First (3x - 2y)(x - 3y) Outer Inner Last 3x2- 9xy- 2xy+ 6y2 Note sum of O + I terms

  16. What to do How to do it 120 15 + 8 = 23 Example: 6t2 + 23t + 20 1. Find the product of 6 and 20: GN: 6 ·20 = 120 2. Find all of the pairs of factors: r and s 120 · 1 24 · 5 20 · 6 60 · 2 40 · 3 15 · 8 with the sum = 23. 30 · 4 12 · 10 3. Middle sign+ is same sign: +15 , + 8 4. Separate middle term23t: +15t , + 8t

  17. What to do How to do it 6t2+ 15t + 8t - 20 Example: 6t2 + 23t + 20 6t2 + 23t + 20 5. Copy the trinomial: 6. Rewrite middle term23t: and group for factoring 7. Factor each group:bring down middle sign 3t(2t + 5) +4(2t + 5) (3t + 4)(2t + 5) 8. Factor common factor:

  18. What to Do How to Do It + 8t +15t 6t2 +20 6t2+ 23t + 20 = (3t + 4)(2t + 5) Check Factors using FOIL Check by multiplying back using F0I L First (3t + 4)(2t + 5) Outer Inner 6t2+ 15t+8t + 20 Last Note sum of O + I terms

  19. What to Do How to Do It Trinomials with Common Factors: Ax2 Bx  C 1. Factor out the common factor(s) from each term. k·ax2k·bxk·c 2. Apply the distributive property. k·(ax2 bx c) 3. As common factorsnumbers are left in composite formandletters are left in power form. ax2 bx c 4. CheckInner Polynomial forClue of Signsand GN

  20. What to Do How to Do It Trinomials with Common Factors: 12x2y - 33xy + 9y 1. Factor out the common factor(s) from each term. 3y·4x2-3y·11x+3y·3 2. Apply the distributive property. 3y(4x2- 11x + 3) 3. As common factorsnumbers are left in composite formandletters are left in power form. 4x2- 11x + 3 4. CheckInner Polynomial forClue of Signsand GN

  21. What to do How to do it 12 - 1 = 11 Inner Trinomial: 4x2 - 11x - 3 1. Find the product of 4 and 3: 4 · 3 = 12 2. Find all of the pairs of factors: r and s 12 = 12 · 1 6 · 2 with the difference = 11. 4 · 3 3. Middle sign– is larger sign: - 12 , + 1 - 12x , + 1x 4. Separate middle term- 11x:

  22. What to do How to do it 4x2- 12x+ 1x - 3 Inner Trinomial: 4x2 - 11x - 3 4x2 - 11x - 3 5. Copy the trinomial: 6. Rewrite middle term-11x: and group for factoring 7. Factor each group:bring down middle sign 4x(x - 3)+1(x - 3) (4x + 1)(x - 3) 8. Factor common factor: Complete: Multiply common factor3y  3y(4x + 1)(x - 3)

  23. What to Do How to Do It + 1x -12x 4x2 - 3 Check Factors by FOIL F0IL  Check factors of inner trinomial by First (4x + 1)(x - 3) Outer Inner Last 4x2- 12x+ 1x- 3 4x2- 11x - 3 Find the sum of O + I terms 12x2y- 33xy - 9y Now, multiply by common factor3y = 3y(4x + 1)(x - 3)

  24. THE END 5.3

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