Math 8H Algebra 1 Glencoe McGraw-Hill JoAnn Evans 8-6 Perfect Squares and Factoring

(n + 6) 2 means It does NOT mean n !! (n + 6) (n + 6) = n 2 + 6n + 6n + 36 The two middle terms are identical. When like terms are combined the result is a perfect square trinomial. = n n + 36 The first term is the square of n, the last term is the square of 6, and the middle term is twice the product of n and 6. When a binomial is squared, the result is a perfect square trinomial. (n + 6)(n + 6)

(x - 5) 2 means It does NOT mean x !! (x - 5) (x - 5) = x 2 - 5x - 5x + 25 The two middle terms are identical. When like terms are combined the result is a perfect square trinomial. = x x + 25 The first term is the square of x, the last term is the square of -5, and the middle term is twice the product of x and -5. When a binomial is squared, the result is a perfect square trinomial. (x – 5)(x - 5)

y y + 49Is this a perfect square trinomial? y 2 is the square of y. 49 is the square of 7. 14y is twice the product of y and 7. y y + 49 will factor as: (y + 7)(y + 7), which is (y + 7) 2.,it’s a perfect square trinomial. When factoring trinomials, look to see if the trinomial fits the perfect square trinomial pattern. If so, you can easily factor it into the square of a binomial.

x 2 – 16x + 64 Is this a perfect square trinomial? x 2 is the square of x. 64 is the square of x is twice the product of x and -8. x 2 – 16x + 64 will factor as: (x - 8)(x - 8), which is (x - 8) 2.,it’s a perfect square trinomial. Why -8 and not 8? Look at the sign on the middle term.

4x x + 25 Is this a perfect square trinomial? 4x 2 is the square of 2x. 25 is the square of 5. 20x is twice the product of 2x and 5. 4x x + 25 will factor as: (2x + 5)(2x + 5), which is (2x + 5) 2.,it’s a perfect square trinomial.

h + h 2 Is this a perfect square trinomial? h 2 is the square of h. 100 is the square of h is twice the product of h and -10. (h - 10)(h - 10) = (h - 10) 2.,it’s a perfect square trinomial. h h + 100

m 2 – 10m - 25 Is this a perfect square trinomial? m 2 is the square of m. -25 isn’t the square of 5 or -5., it’s not a perfect square trinomial.

Factor fully: 3x 3 + 6x 2 + 3x This doesn’t appear to fit the pattern of a perfect square trinomial. How about checking first for a GCMF? There’s a GCMF of 3x. 3x (x 2 + 2x + 1) 3x 3 + 6x 2 + 3x 3x (x + 1) (x + 1) 3x (x + 1) 2 x 2 is the square of x. 1 is the square of 1. 2x is twice the product of x and 1. Now it’s fully factored.

Even if you don’t recognize that a trinomial is a perfect square, you can still factor it using the X or the X-Box method. Recognition of perfect square trinomials is a convenient (although not essential) help when factoring. Here are two of the PSTs we worked with earlier. x 2 – 16x ● (x – 8)(x – 8) 4x x ● (2x + 5)(2x + 5) 4x x 2x

HOW TO APPROACH FACTORING PROBLEMS When factoring it’s important to approach the problem in the following order. Look at the problem carefully and ask yourself: 1.Is there a GCMF? If so, factor it out; but then take another look to check for opportunities to factor again. 2. Is this the DIFFERENCE of Squares? Remember that these never have a middle (x) term. If so, factor it into the ( + )( – ) pattern. 3. Is this a Perfect Square Trinomial? If so, it will factor into two binomials that are exactly the same and can be written as a square.

Check by doing FOIL. Remember that factoring is working backwards from multiplying. Always look at what remains and check for opportunities to factor again. Don’t give up too soon! 4.If the problem isn’t the difference of squares or a PST, you must factor using the X. Remember to use the X-Box if the leading coefficient is more than 1. 5.If there are four or more terms, try the Factor by Grouping method. Group terms with common factors and then factor out the GCF from each grouping. Then use the distributive property a second time to factor a common binomial factor.

Factor Fully 6m 4 – 12m (m 2 - 1)(m 2 – 1) 6(m + 1) 2 (m - 1) 2 6(m 4 – 2m 2 + 1) GCMF? perfect square trinomial difference of squares 6(m + 1)(m – 1)(m + 1) ( m – 1)

3q 3 – 60q q 3q(q – 2)(q - 18) x 2 y + 36xy + 108y 3y(x + 6) 2 3y(x x + 36) Factor Fully GCMF? perfect square trinomial GCMF? 3q(q 2 – 20q + 36) not a perfect square trinomial—use the X

Solve by Factoring 5y 3 = 20y 5y y = 0 5y(y 2 – 4) = 0 5y(y + 2)(y – 2) = 0 5y = 0 or y + 2 = 0 or y – 2 = 0 y = 0; -2; 2 GCMF? Difference of squares

Solve by Factoring 8x x – 60 = 0 4(2x 2 – 7x – 15) = 0 4(2x + 3)(x – 5) = 0 2x + 3 = 0 or x - 5 = 0 + ● x 2 3x 2x3 x x5 x = ; 5 GCMF? X-Box

You’ve solved equations like x 2 – 36 = 0 by factoring. You can also solve them using the definition of a square root. For any number n > 0, if x 2 = n, then x = ± Think of it this way. When solving equations, whatever you do to one side of the equation, you must do to the other side of the equation. If you unsquare one side, you must also unsquare the other side of the equation.

You can also use the square root property when you have a squared binomial. This represents two separate solutions. Solve each equation.

Solve Using the Square Root Property

5 is not a perfect square.