Factoring by Grouping

Factoring by grouping is a technique used to factor a polynomial with four terms by grouping the terms into pairs and factoring out a common factor from each pair.

Steps to Factor by Grouping:

Group the terms: Group the four terms of the polynomial into pairs.
Factor out the common factor: Factor out the greatest common factor from each pair of terms.
Factor out the GCF of the entire expression: Factor out the greatest common factor from the resulting two terms.
Check for common factors: Check to see if there are any common factors that can be factored out from the entire expression.

Example:

Factor the polynomial 2x² + 8x + 3x + 12

Solution:

Step 1: Group the terms

2x² + 8x + 3x + 12

(2x² + 8x) + (3x + 12)

Step 2: Factor out the common factor

2x(x + 4) + 3(x + 4)

Step 3: Factor out the GCF of the entire expression

(x + 4)(2x + 3)

So, the factored form of the given polynomial is (x + 4)(2x + 3).

Study Guide:

When factoring by grouping, it's important to first identify if the given polynomial has four terms. If it does, then proceed with the steps for factoring by grouping. Remember to look for common factors in the pairs of terms and the entire expression to ensure that the polynomial is fully factored.

Practice factoring by grouping with different polynomials to strengthen your understanding of the technique.

Remember to always check your factored form by multiplying the factors to ensure that the result is the original polynomial.

Good luck with your factoring by grouping practice!

◂Math Worksheets and Study Guides Eighth Grade. Collecting and describing data

Study Guide

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

Worksheet/Answer key

Collecting and describing data

The resources above cover the following skills:

Data Analysis and Probability (NCTM)

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.

Select and use appropriate statistical methods to analyze data.

Find, use, and interpret measures of center and spread, including mean and interquartile range.

Develop and evaluate inferences and predictions that are based on data.

Use conjectures to formulate new questions and plan new studies to answer them.

Grade 8 Curriculum Focal Points (NCTM)

Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets

Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.

Connections to the Grade 8 Focal Points (NCTM)

Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.