Sample Lessons and Worksheets

In this grade MINDsprinting's math curriculum covers:

• Introduction to algebra
• Addition, subtraction, multiplication, and division of polynomials
• Factoring polynomials

Plus any other topics your child needs to work on, as determined by our FREE Assessment Test. You can view a complete list of the topics covered in our curriculum by clicking here.

Below is a free sample worksheet from our 11th grade math curriculum. It is only one of many worksheets that are automatically assigned to students in this grade level based on their Assessment Test results and their progress through our curriculum. Each worksheet is designed to take between 10 and 20 minutes to complete. We recommend students do at least 3 worksheets per week, though you can complete as many as you want using the MINDsprinting system.

The lesson below is a HTML reproduction of our worksheet. All lessons are served in PDF format and are designed to be printed out and completed by hand.

Lesson M87-1 - Factoring Type 1 Polynomials - 1

When we multiply two polynomials together we can write the product in expanded form. If we were to write an expanded polynomial as the product of two (or more) polynomials, we would be putting the polynomial into factored form.

factored form -->   (x + 2)(x + 3) = x2+5x + 6   <-- expanded form

To factor means to rewrite a polynomial as a product of polynomials. In the above example (x + 2) and (x + 3) are the factors of (x2+ 5x + 6).

How do we factor a trinomial like x2+7x + 12?

Recall our general expansion:

(x + a)(x + b) = x2 + (a + b)x + ab

The sum of a and b gives us the coefficient of the x term and the product of a and b gives us the constant term.

In the example x2+7x + 12 we need to find two numbers such that their sum is +7 and their product is +12.

Start by listing the factors of 12 in a chart, along with the sum of the factors.

 Factors Sum 1, 12 13 2, 6 8 3, 4 7 -3, -4 -7 -2, -6 -8 -1, -12 -13

We can see from the chart that the only pair of numbers that have a product or +12 and a sum of +7 are 3 and 4.

Thus we can factor as follows:

x2+ 7x + 12 = (x + 3)(x + 4)

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 Factors of -6 Sum 1, -6 -5 2, -3 -1 -2, 3 1 -1, 6 5

Example 1: Factor x2+ x - 6.

We need to find a pair of numbers that sum to +1 and have a product of -6.

x2+ x - 6 = (x - 2)(x + 3)

 Factors of 9 Sum 1, 9 10 3, 3 6 -3, -3 -6 -1, -9 -10

Example 2: Factor x2- 6x +9.

Sum must = -6
Product must = +9

(x2- 6x + 9) = (x - 3)(x - 3)
= (x - 3)2

Solve the following.

 1. (x - 21)(x +6) 2. (x + 18)(x - 10) 3. (x + 7)(x - 14) 4. (x +5)(x + 19)

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