Factoring a Polynomial

1. Factor out any common factors.2. Factor according to one of the

special polynomial forms. 3. Factor as ax2 + bx + c = (mx + r)

(nx + s). 4. Factor by grouping.

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.

Try this on your own.

x3 – 9x x(x2 – 9) x · (x + 3)(x – 3)

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)

Try this on your own.

25 – x2

u = 5, v = x (5 + x)(5 – x)

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

Try this on your own.

x2 + 10x + 25 u = x, v = 5 (x + 5)2

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*

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)

Try this on your own.

y3 – 8 u = y, v = 2 (y – 2)(y2 + 2y + 4)

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)

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.

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)

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)

Try this one on your own.

x2 – 2x – 35 (x – 7)(x + 5)

Assignment pg. 42

#7-10 TOM 3 #11-16 TOM 4 #17-22 TOM 4 #27-34 TOM 6

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.

Pg. 42 #35 – 40 TOM = 4

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.

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)

Factoring by grouping.

Pg. 42#47-52 TOM = 4