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Factoring a Polynomial

Factoring a Polynomial. Factor out any common factors. Factor according to one of the special polynomial forms. Factor as ax 2 + bx + c = (mx + r)(nx + s). Factor by grouping. Removing a Common Factor. 3 – 12x 2 Find the greatest common factor of each term. 3

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Factoring a Polynomial

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  1. Factoring a Polynomial • Factor out any common factors. • Factor according to one of the special polynomial forms. • Factor as ax2 + bx + c = (mx + r)(nx + s). • Factor by grouping.

  2. Removing a Common Factor • 3 – 12x2 • Find the greatest common factor of each term. • 3 • Divide each term by the greatest common factor. • 3(1 – 4x2) • Continue factoring (use special products if possible) • 3 · (1 – 2x)(1 + 2x) – from sum and difference of same terms.

  3. Try this on your own. • x3 – 9x • x(x2 – 9) • x · (x + 3)(x – 3)

  4. Factoring the Difference of Two Squares • (x + 2)2 – y2 • Identify first term (u) and second term (v). • u = (x + 2), v = y • Use the difference of two squares form. (u + v)(u – v) • [(x + 2) + y][(x + 2) – y] • Simplify • (x + 2 + y)(x + 2 – y)

  5. Try this on your own. • 25 – x2 • u = 5, v = x • (5 + x)(5 – x)

  6. Perfect Square Trinomials • A perfect square trinomial is the square of a binomial. • Identify a perfect square trinomial • First and last terms are squares and the middle is 2uv. • 16x2 + 8x + 1 (16x2 and 1 are squares of 4x and 1, the middle equals 2·4x·1) • Identify u and v • u = 4x, v = 1 • Rewrite as the square of (u + v) • (4x + 1)2

  7. Try this on your own. • x2 + 10x + 25 • u = x, v = 5 • (x + 5)2

  8. With a minus sign after the first term. • 9x2 – 12x + 4 • u = 3x, v = 2 • (3x – 2)2 *Note that the only change is the sign of your answer*

  9. Factoring the Difference of Cubes • x3 – 27 • Identify u and v • u = x, v = 3 • Rewrite using difference of cubes form. (u – v)(u2 + uv + v2) • (x - 3)(x2 + 3x + 9)

  10. Try this on your own. • y3 – 8 • u = y, v = 2 • (y – 2)(y2 + 2y + 4)

  11. Factoring the Sum of Cubes • x3 + 64 • Identify u and v • u = x, v = 4 • Rewrite using sum of cubes form. (u + v)(u2 - uv + v2) • (x + 4)(x2 – 4x + 16)

  12. Factoring a Trinomial when the leading coefficient is 1 • A few tricks. • The sign between the second and third term determine if the signs in the binomial are the same or different. + means the same, - means different.

  13. Factoring a Trinomial when the leading coefficient is 1 • Ex: x2 – 7x + 12 • *The plus tells us the signs of the binomial factors will be the same.* The minus sign between the first and second terms tells us the both will be -. • To factor we think of factors of 12 that add to 7. • (x – 4)(x – 3)

  14. Factoring a Trinomial when the leading coefficient is 1 • Ex: x2 – 5x – 6 • Since the sign between the second and third term is -, the signs of the binomials will be different. The – sign between the first and second term tells us the bigger of the binomial factors is -. • (x – 6)(x + 1)

  15. Try this one on your own. • x2 – 2x – 35 • (x – 7)(x + 5)

  16. Assignment pg. 42 • #7-10 TOM 3 • #11-16 TOM 4 • #17-22 TOM 4 • #27-34 TOM 6

  17. Factoring a trinomial when the leading coefficient is not 1. • 2x2 + x – 15 • List all the possible factorizations. • (2x + 15)(x – 1) (2x – 15)(x + 1) (2x + 5)(x – 3) (2x – 5)(x + 3) (2x + 3)(x – 5) (2x – 3)(x + 5) (2x + 1)(2x – 15) (2x – 1)(2x + 15) • Test to find the middle term. • (2x + 5)(x – 3) = 2x2 + 5x – 6x – 6 = 2x2 – x – 6 • It is trial and error.

  18. Pg. 42 #35 – 40 TOM = 4

  19. Factoring by grouping. • Factoring by grouping • Used for polynomials that can not be factored with other methods. • Especially useful when the polynomial has more than three terms.

  20. Factoring by grouping. • X3 – 2x2 – 3x + 6 • Use parenthesis to group. • (x3 – 2x2) - (3x + 6) • Remove the common factors. • x2(x – 2) – 3(x – 2) • Rewrite using the distributive property. • (x - 2) (x2 – 3)

  21. Factoring by grouping. • Pg. 42 #47-52 TOM = 4

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