MA.912.AR.1.7

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.NSO.1.1
• MA.912.AR.3.1
• MA.912.AR.3.5
• MA.912.AR.3.6
• MA.912.AR.3.7
• MA.912.AR.3.8

Terms from the K-12 Glossary

• Expression  
• Polynomial

Vertical Alignment

Previous Benchmarks

• MA.8.AR.1.3

Next Benchmarks

• MA.912.AR.1.8

Purpose and Instructional Strategies

• 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) 

• 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). 

• Box Method 
  • To factor $a$$x$² + $b$$x$ + $c$  the general box method is shown below.

• 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.

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).

Instructional Tasks

• 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? 

• 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.

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

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

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

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

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