### Examples

*Example:*The expression is equivalent to the factored form .

*Example:* The expression is equivalent to the factored form
.

### Clarifications

*Clarification 1:*Within the Algebra 1 course, polynomial expressions are limited to 4 or fewer terms with integer coefficients.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Expression
- Polynomial

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students rewrote binomial algebraic expressions as a common factor times a binomial. In Algebra I, students rewrite polynomials, up to 4 terms, as a product of polynomials over the real numbers. In later grades, students will rewrite polynomials as a product of polynomials over the real and complex number systems.- Instruction includes special cases such as difference of squares and perfect square trinomials.
- Instruction builds upon student prior knowledge of factors, including greatest common factors.
- Instruction includes the student understanding that factoring is the inverse of multiplying polynomial expressions.
- Instruction includes the use of models, manipulatives and recognizing patterns when
factoring.
o
- Sum-Product Pattern

- The expression $x$
^{2 }+ 7$x$ + 10 can be written as ($x$ + 5)($x$ + 2) since 5 + 2 = 7 and 5(2) = 10. - Factor by Grouping
- The expression $x$
^{3}+ 7$x$^{2}+ 2$x$ + 14 can be grouped into two binomials and rewritten as ($x$^{3}+ 7$x$^{2}) + (2$x$ + 14). Each binomial can be factored and rewritten as $x$^{2}($x$ + 7) + 2($x$ + 7) resulting in the same factor and the factored form as ($x$^{2}+ 2)($x$ + 7)

- The expression $x$

- A-C Method
- When factoring trinomials $a$$x$
^{2}+ $b$$x$+ $c$ , multiply $a$ and $c$, then determine factor pairs of the product. Using the factor pair that add to $b$ and multiply to $c$, rewrite the middle term and then factor by grouping.- For example, given 2$x$
^{2}+ $x$ − 6 and that $a$$c$ = −12, one can determine that two numbers that add to 1 and multiple to -12 are 4 and -3. This information can be used to rewrite the given quadratic as 2$x$^{2}+ 4$x$ − 3$x$ − 6 . Then, using factor by grouping the expression is equivalent to (2$x$^{2}+ 4$x$) - (3$x$ + 6) which is equivalent to 2$x$($x$ + 2) − 3($x$ + 2) which is equivalent to the factored form (2$x$ − 3)($x$ + 2).

- For example, given 2$x$

- When factoring trinomials $a$$x$

- Box Method
- To factor $a$$x$
^{2}+ $b$$x$ + $c$ the general box method is shown below.

- To factor $a$$x$

- For example, to factor 2$x$
^{2}- 9$x$ - 5 the box method is shown below.

- Area Model (Algebra tiles)
- The factorization of 2$x$
^{2}- 9$x$ - 5 using algebra tiles is shown below.

- The factorization of 2$x$

### Common Misconceptions or Errors

- Students may not identify the greatest common factor or factor completely.

### Strategies to Support Tiered Instruction

- Instruction includes providing a flow chart to reference while completing examples.
- Instruction includes providing definition of greatest common factor and strategies for
identifying the greatest common factor of numerical or algebraic terms.
- For example, the expression 8$x$
^{3}- 4$x$^{2}has common factors of 2 and $x$, but these are not greatest common factors. The greatest common factor of the coefficients is 4 and the greatest common factor of the variable terms is $x$^{2}. So, the greatest common factor of the two terms is 4$x$^{2}. The expression 8$x$^{3}- 4$x$^{2}can be rewritten as 4$x$^{2}(2$x$ -1).

- For example, the expression 8$x$

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.4.1, MTR.5.1)*

- Part A. Given the polynomial $x$
^{4}– 16$y$^{4}$z$^{8}, rewrite it as a product of polynomials.- Part B. Discuss with your partner the strategy used. How do your polynomial factors compare to one another?

Instructional Task 2 (MTR.3.1,

Instructional Task 2 (MTR.3.1,

*MTR.5.1*)

- Part A. What are the factors of the quadratic 16$x$
^{2 }- 48$x$ + 36?- Part B. Determine the roots of the quadratic function $f$($x$) = 16$x$
^{2 }- 48$x$ + 36.- Part C. What do you notice about your answers from Part A and Part B?
- Part D. Graph the function $f$($x$) = 16$x$
^{2 }- 48$x$ + 36.

### Instructional Items

*Instructional Item 1*

- Given the polynomial $x$
^{4}– 16$y$^{4}$z$^{8}, rewrite it as a product of polynomials.

*Instructional Item 2*

- Given the polynomial $x$
^{2 }- 10$x$ + 24, rewrite it as a product of polynomials.

*Instructional Item 3*

- Given the polynomial $x$
^{3}- 3$x$^{2 }- 9$x$ + 27 rewrite it as a product of polynomials.

*Instructional Item 4*

- What is one of the factors of the polynomial 21$r$
^{3 }$s$^{2 }- 14 $r$^{2}$s$?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Teaching Idea

## MFAS Formative Assessments

Students are asked to find the width of a rectangle whose area and length are given as polynomials.

Students are asked to rewrite quadratic expressions and identify parts of the expressions.

Students are asked to identify equivalent quadratic expressions and to name the form in which each expression is written.

Students are asked to rewrite numerical expressions to find efficient ways to calculate.

## Original Student Tutorials Mathematics - Grades 9-12

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2: Factoring Polynomials Using Special Cases (Current Tutorial)
- Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

Learn to factor quadratic trinomials when the coefficient *a* does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4: Factoring Polynomials when
*a*Does Not Equal 1: Snowflake Method (Current Tutorial) - Part 5:
**Multistep Factoring: Quadratics**

Learn how to factor polynomials by finding their greatest common factor in this interactive tutorial.

Learn how to factor quadratic polynomials when the leading coefficient (*a*) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method (Current Tutorial)
- Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5: Multistep Factoring: Quadratics (current tutorial)

Learn how to factor quadratics when the coefficient *a* = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

- Part 1: The Diamond Game: Factoring Quadratics when
*a*= 1 (Current Tutorial) - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

## Student Resources

## Original Student Tutorials

Learn how to factor polynomials by finding their greatest common factor in this interactive tutorial.

Type: Original Student Tutorial

Learn how to use multistep factoring to factor quadratics in this interactive tutorial.

This is part 5 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5: Multistep Factoring: Quadratics (current tutorial)

Type: Original Student Tutorial

Learn to factor quadratic trinomials when the coefficient *a* does not equal 1 by using the Snowflake Method in this interactive tutorial.

This is part 4 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4: Factoring Polynomials when
*a*Does Not Equal 1: Snowflake Method (Current Tutorial) - Part 5:
**Multistep Factoring: Quadratics**

Type: Original Student Tutorial

Learn how to factor quadratic polynomials when the leading coefficient (*a*) is not 1 by using the box method in this interactive tutorial.

This is part 3 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method (Current Tutorial)
- Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

Type: Original Student Tutorial

Learn how to factor quadratics when the coefficient *a* = 1 using the diamond method in this game show-themed, interactive tutorial.

This is part 1 in a five-part series. Click below to open the other tutorials in this series.

- Part 1: The Diamond Game: Factoring Quadratics when
*a*= 1 (Current Tutorial) - Part 2:
**Factoring Polynomials Using Special Cases** - Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

Type: Original Student Tutorial

Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.

This is part 2 in a five-part series. Click below to open the other tutorials in this series.

- Part 1:
**The Diamond Game: Factoring Quadratics when***a*= 1 - Part 2: Factoring Polynomials Using Special Cases (Current Tutorial)
- Part 3:
**Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method** - Part 4:
**Factoring Polynomials when***a*Does Not Equal 1: Snowflake Method - Part 5:
**Multistep Factoring: Quadratics**

Type: Original Student Tutorial