Math 8H Algebra 1 Glencoe McGraw-Hill JoAnn Evans 8-4 Factoring Trinomials ax 2 + bx + c.

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Math 8H Algebra 1 Glencoe McGraw-Hill JoAnn Evans 8-4 Factoring Trinomials ax 2 + bx + c

+ ● ax 2 c To complete the process, you’ll need to construct a 4-square box. Quadratic trinomials that have an “a” value other than 1 require an extra step beyond the factoring X.

+ ● Put a∙c in the top of the X. 2 ∙ -5 = Put b in the bottom of the X. -3 a = 2 b = -3 c = Name two numbers that have a product of -10 and a sum of -3. Factor: 2x 2 – 3x - 5

+ ● x 2 – 3x - 5 2x x 2x Put ax 2 in the top left corner. Put c in the lower right corner. Put the numbers from the sides of the X figure in the two remaining boxes. Their order is not important. Add an “x” after each of the side numbers. Notice that all terms of the trinomial are represented in the box. The ax 2 and the c appear as they do in the original trinomial. The middle term of the original trinomial is the sum of the remaining squares.

2x x 2x Find the GCMF in each column. Write it above the column. Always pull out only the positive common factors. x1 Find the GCMF in each row. Write it beside the row. Always pull out only the positive common factors. 2x 5 The sign between the two terms will be the sign of the coefficient in the diagonal boxes. + - (2x – 5)(x + 1) is the factored form. 2x 2 – 3x - 5

+ ● x x + 4 Put ax 2 in the top left corner. Put c in the lower right corner. Put the side numbers in the two remaining boxes with an “x” after them. Their order is not important. 6x 2 43x 8x 3x4 2x Find the GCMF in each row and each column. (3x + 4)(2x + 1) is the factored form. Check your solution!

+ ● x x - 12 Put ax 2 in the top left corner. Put c in the lower right corner. Put the side numbers in the two remaining boxes with an “x” after them. Their order is not important. 5x x 2x 5x2 x Find the GCMF in each row and each column. (5x + 2)(x - 6) is the factored form. Check your solution!

+ ● x 2 - 5x - 1 Put ax 2 in the top left corner. Put c in the lower right corner. Put the side numbers in the two remaining boxes with an “x” after them. Their order is not important. 14x 2 -7x 2x 7x1 2x Find the GCMF in each row and each column. (7x + 1)(2x - 1) is the factored form. Check your solution!

+ ● x x + 6 Put ax 2 in the top left corner. Put c in the lower right corner. Put the side numbers in the two remaining boxes with an “x” after them. Their order is not important. 5x 2 610x 3x 5x3 x Find the GCMF in each row and each column. (5x + 3)(x + 2) is the factored form. Check your solution!

On your own: 6x 2 + x - 122x 2 - 7x ● ● x x 9x 2x x x 2x x x 2x (2x + 3)(3x - 4)(x + 1)(2x - 9)

Caution! This method will NOT work if the trinomial has a GCMF that could be factored out first. Example: 6x 2 – 8x - 8 Factor out the GCMF of 2 before proceeding with the X-Box method! + ● x x 2x 3x2 2 x + - 2(3x 2 – 4x – 4) a = 3 b = -4 c = -4 2(3x + 2)(x - 2) is the factored form. Don’t forget to include the GCMF that was factored out first!

Caution! Look for a GCMF that can be factored out first! 2x 2 y + 10xy – 72y What is the GCMF of these three terms? + ● y(x 2 + 5x – 36) a = 1 b = 5 c = -36 2y(x - 4)(x + 9) is the factored form. Don’t forget to include the GCMF that was factored out first! 2y 2y(x 2 + 5x – 36)

If there are 4 or more terms, factor by the grouping method. 10x 2 - 8x - 35x + 28 The first two terms have a common factor of 2x. What common factor do the second two terms have? -7 Factor out the GCMF from each pair of terms. Write the expression in factored form. 2x(5x – 4) (2x – 7)(5x – 4) -7(5x – 4)

If there are 4 or more terms, factor by the grouping method. 3x 3 - 5x 2 + 9x - 15 The first two terms have a common factor of x 2. What common factor do the second two terms have? 3 Factor out the GCMF from each pair of terms. Write the expression in factored form. x 2 (3x – 5) + 3(3x – 5) (x 2 + 3)(3x – 5)

Solve Equations by Factoring 3x 2 – 5x = 12 3x 2 – 5x – 12 = 0 + ● x x 4x 3x4 3 x + - (3x + 4)(x – 3) = 0 3x + 4 = 0 or x – 3 = 0 x = Collect all terms on the left hand side of the equation. Factor the left side using the X-Box method. Use the Zero Product Property.