Using Diagrams to Understand Equivalent Expressions

8.6.2 Using Diagrams to Understand Equivalent Expressions

8.6.2 • Using Diagrams to Understand Equivalent Expressions

Activity

Look at these two expressions.

$(x + 7) (x + 9)$

$x^{2} + 16 x + 63$

We can show they are equivalent by using a diagram to break apart the different pieces.

	$x$	7
$x$	$x^{2}$	$7 x$
9	$9 x$	63

Multiplication grid illustrating the distributive property for the product of (x + 9) and (x + 7).

The binomial factors $(x + 7)$ and $(x + 9)$ have been placed in the cells along the horizontal and vertical edge of the diagram.

To find the internal cells, the terms in each factor are multiplied together.

Then, the like terms in the product cells are combined. Notice that $7 x + 9 x$ results in the linear term of our second expression, $16 x$ (from $x^{2} + 16 x + 63$ ).

Use a diagram to show that each pair of expressions is equivalent. It is helpful to rewrite any subtraction expressions as adding the opposite before starting, as has been modeled in questions 2 and 5.

$x (x + 3)$ and $x^{2} + 3 x$

Compare your answer:

$x (x + 3)$ and $x^{2} + 3 x$

	$x$	0
$x$	$x^{2}$	0
3	$3 x$	0

This table illustrates the multiplication of base values (x, 3) by factors 'x' and '0', demonstrating algebraic products and the property of multiplication by zero.

$x (x + (- 6))$ and $x^{2} - 6 x$

Compare your answer:

$x (x + (- 6))$ and $x^{2} - 6 x$

	$x$	0
$x$	$x^{2}$	0
$- 6$	$- 6 x$	0

Area model demonstrating polynomial multiplication, specifically (x-6) by (x+0), showing individual term products like x^2 and -6x.

$(x + 2) (x + 4)$ and $x^{2} + 6 x + 8$

Compare your answer:

$(x + 2) (x + 4)$ and $x^{2} + 6 x + 8$

	$x$	2
$x$	$x^{2}$	$2 x$
4	$4 x$	8

A multiplication grid illustrating products of algebraic expressions and integers.

$(x + 4) (x + 10)$ and $x^{2} + 14 x + 40$

Compare your answer:

$(x + 4) (x + 10)$ and $x^{2} + 14 x + 40$

	$x$	4
$x$	$x^{2}$	$4 x$
10	$10 x$	40

Multiplication grid for (x + 10)(x + 4) showing individual term products.

$(x + (- 5)) (x + (- 1))$ and $x^{2} - 6 x + 5$

Compare your answer:

$(x + (- 5)) (x + (- 1))$ and $x^{2} - 6 x + 5$

	$x$	–5
$x$	$x^{2}$	$- 5 x$
–1	$- 1 x$	5

An area model displaying the individual terms generated from the binomial multiplication of (x-1) and (x-5).

$(x - 1) (x - 7)$ and $x^{2} - 8 x + 7$

Compare your answer:

$(x - 1) (x - 7)$ and $x^{2} - 8 x + 7$

Rewrite $(x - 1) (x - 7)$ as $(x + (- 1)) (x + (- 7))$ .

	$x$	–1
$x$	$x^{2}$	$- 1 x$
–7	$- 7 x$	7

This grid illustrates the multiplication of binomials (x - 7) and (x - 1), showing the individual term products.

Review the pairs of expressions that involve the product of two sums or two differences that you just diagramed. How is each expression in factored form (the first expression placed into the cells along the horizontal and vertical edge of the diagram) related to the equivalent expression in standard form (the second expression that results from the product cells)?

Compare your answer:

Because the coefficients of the $x$ -terms are both 1 in the factored form, when the two numbers that appear in factored form are added, the result is the coefficient of $x$ in standard form. If those same two numbers are multiplied, the result is the constant term in standard form. (Note that this explanation relies on recasting every subtraction expression as adding the opposite.)

Self Check

An expression in standard form is shown. The diagram below is equivalent to which expression in factored form?

$x^2−13x + 42$

	$x$	-6
$x$	$x^2$	$-6x$
-7	$-7x$	42

$(x - 6)(x - 7)$
$(x - 6)(x + 7)$
$(x + 6)(x + 7)$
$(x - 13)(x + 42)$

Additional Resources

Using Diagrams to Understand Equivalent Expressions

Example 1

Look at these two expressions.

$(x + 3) (x + 5)$

$x^{2} + 8 x + 15$

We can show they are equivalent by using a diagram to break apart the different pieces.

Step 1 - First, let’s populate the diagram.

Diagram that shows multiplication of x plus 5 times x plus 3 times x and 5 are on the left and x and 3 are on the top there are 4 sections to be filled in.

Step 2 - Then we multiply the corresponding rows and columns.

	$x$	3
$x$	$x \cdot x = x^{2}$	$3 \cdot x = 3 x$
5	$5 \cdot x = 5 x$	$3 \cdot 5 = 15$

Multiplication grid showing products of variable 'x' and constants '3' and '5'.

This leaves the diagram shown here.

	$x$	3
$x$	$x^{2}$	$3 x$
5	$5 x$	15

Multiplication grid illustrating the expansion of (x+5)(x+3) by showing individual product terms.

Step 3 - We can use this diagram to find our expression in standard form.

Diagram that represents multiplication of x plus 5 times x plus 3 times x and 5 are on the left and x and 3 are on the top the 4 sections are filled in as x-squared, 3 times x, 5 times x, and 15 the product of x-squared plus 8 times x plus 15 is shown, with a green arrow from x-squared in the diagram to x-squared in the expression, two blue arrows from 3 times x and 5 times x in the diagram to 8 times x in the expression, and a red arrow from 15 in the diagram to 15 in the expression.

So, the expressions are equivalent.

Example 2

The same can be done with two factors using subtraction as well.

$(x - 6) (x - 9)$

$x^{2} - 15 x + 54$

It is helpful to rewrite the factored form as the addition of negative numbers when subtracting occurs in the factors.

So $(x - 6) (x - 9)$ becomes $(x + (- 6)) (x + (- 9))$ . Now the process is the same.

Step 1 - Let’s populate the table.

	$x$	–6
$x$
–9

An algebraic grid template showing 'x' and '-6' as column headers and 'x' and '-9' as row headers, likely for algebraic calculations.

Step 2 - Multiply the columns and rows.

	$x$	–6
$x$	$x^{2}$	$- 6 x$
–9	$- 9 x$	54

Multiplication grid illustrating the product of (x - 9) and (x - 6) using a visual method.

Step 3 - Use the diagram to write the expression in standard form.

$x^{2} - 15 x + 54$

The expressions are equivalent.

When the coefficients of the $x$ -terms are both 1 in the factored form:

Take notice that the linear term ( $x$ -term) in the expression in standard form is the sum of the two numbers in the expression in factored form.

In the example above, the term $- 15 x$ is the sum of the two numbers –9 and –6.

Also, the constant term is the product of the two numbers in the expression in factored form.

In the example above, the term 54 is the product of the two numbers –9 and –6.

Recognizing this will be helpful for switching between standard and factored forms. Remember, this pattern only occurs when the coefficients of the $x$ -terms in the factors are 1 (because they result in a quadratic term that has a coefficient of 1 as well, $x^{2}$ ).

Try it

Using Diagrams to Understand Equivalent Expressions

1. Use a diagram to write $(x + 2) (x + 7)$ in standard form.

	$x$	–3
$x$	$x^{2}$	$- 3 x$
–8	$- 8 x$	24

This table illustrates the algebraic expansion of the product (x-8)(x-3) using a grid method.

Based on the diagram, what is an equivalent factored form of this expression?

Here is how to use a diagram to help convert between standard and factored forms:

$x^{2} + 9 x + 14$

	$x$	2
$x$	$x^{2}$	$2 x$
7	$7 x$	14

This table presents the products of algebraic terms and numbers in a multiplication grid format.

2. The diagram shown breaks down $x^{2} - 11 x + 24$ .

	$x$	–3
$x$	$x^{2}$	$- 3 x$
–8	$- 8 x$	24

Area model demonstrating the multiplication of (x-8) and (x-3) through individual term products.

Based on the diagram, what is an equivalent factored form of this expression?

Here is how to use a diagram to help convert between standard and factored forms:

$(x - 3) (x - 8)$