240 likes | 586 Views
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.
E N D
BeginningAlgebra 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. Objective 2. To factor a polynomial by first factoring out the greatest common factor and then factoring the polynomial that remains.
Same sign + Sum Larger sign – Difference Clue of Signs Ax2 Bx C Read as +or–
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
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 AC= P 2. Find all of the pairs of factorsr and s P = rs, r > s r + s = B 3. so their sum or difference is the middle coefficient . is the middle coefficient .
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 = AC is the grouping number P = rs , 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)
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
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:
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
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:
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:
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
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:
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:
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
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
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:
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
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·ax2k·bxk·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
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
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:
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)
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)
THE END 5.3