Question Expand the expression $(x+6)(x+5)$ and illustrate the resulting product as a rectangle.

Studdy Solution

STEP 1

Assumptions
1. We are asked to complete a rectangle to illustrate the product of two binomials, $(x+6)$ and $(x+5)$ .
2. The rectangle will be divided into smaller rectangles or squares, each representing a term from the expansion of the product.
3. The length and width of the rectangle will correspond to the terms in the binomials.

STEP 2

To illustrate the product of $(x+6)$ and $(x+5)$ as a rectangle, we will draw a rectangle and divide it into parts that represent each term in the product.

STEP 3

The first term of each binomial, $x$ , will represent one dimension of the rectangle. The second terms, $6$ and $5$ , will represent the other dimension.

STEP 4

Draw a rectangle and label one side with $x+6$ and the adjacent side with $x+5$ .

STEP 5

Divide the rectangle into four smaller parts by drawing a line parallel to the side labeled $x+5$ at a distance of $6$ units from it and another line parallel to the side labeled $x+6$ at a distance of $5$ units from it.

STEP 6

Label the four smaller rectangles/squares that result from the division. The top left square will be $x \times x$ , the top right rectangle will be $x \times 5$ , the bottom left rectangle will be $6 \times x$ , and the bottom right rectangle will be $6 \times 5$ .

STEP 7

Write the area of each smaller rectangle/square inside them, corresponding to the product of their sides. The top left square will have an area of $x^2$ , the top right rectangle will have an area of $5x$ , the bottom left rectangle will have an area of $6x$ , and the bottom right rectangle will have an area of $30$ .

STEP 8

The completed rectangle now illustrates the product $(x+6)(x+5)$ with each part of the rectangle representing a term in the expanded form of the product.

STEP 9

The expanded form of the product can be found by adding the areas of all the smaller rectangles/squares.

STEP 10

Write down the expanded form using the areas of the smaller parts:
$x^2 + 5x + 6x + 30$

STEP 11

Combine like terms in the expanded form to simplify the expression.
$x^2 + (5x + 6x) + 30$

STEP 12

Calculate the sum of the like terms $5x$ and $6x$ .
$5x + 6x = 11x$

STEP 13

Write the final simplified form of the expanded product.
$x^2 + 11x + 30$
The rectangle illustrates the product $(x+6)(x+5)$ and its expansion is $x^2 + 11x + 30$ .

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Question Expand the expression $(x+6)(x+5)$ and illustrate the resulting product as a rectangle.

Studdy Solution

STEP 1

STEP 2

To illustrate the product of $(x+6)$ and $(x+5)$ as a rectangle, we will draw a rectangle and divide it into parts that represent each term in the product.

STEP 3

The first term of each binomial, $x$ , will represent one dimension of the rectangle. The second terms, $6$ and $5$ , will represent the other dimension.

STEP 4

Draw a rectangle and label one side with $x+6$ and the adjacent side with $x+5$ .

STEP 5

Divide the rectangle into four smaller parts by drawing a line parallel to the side labeled $x+5$ at a distance of $6$ units from it and another line parallel to the side labeled $x+6$ at a distance of $5$ units from it.

STEP 6

Label the four smaller rectangles/squares that result from the division. The top left square will be $x \times x$ , the top right rectangle will be $x \times 5$ , the bottom left rectangle will be $6 \times x$ , and the bottom right rectangle will be $6 \times 5$ .

STEP 7

STEP 8

The completed rectangle now illustrates the product $(x+6)(x+5)$ with each part of the rectangle representing a term in the expanded form of the product.

STEP 9

The expanded form of the product can be found by adding the areas of all the smaller rectangles/squares.

STEP 10

Write down the expanded form using the areas of the smaller parts:
$x^2 + 5x + 6x + 30$

STEP 11

Combine like terms in the expanded form to simplify the expression.
$x^2 + (5x + 6x) + 30$

STEP 12

Calculate the sum of the like terms $5x$ and $6x$ .
$5x + 6x = 11x$

STEP 13

Write the final simplified form of the expanded product.
$x^2 + 11x + 30$
The rectangle illustrates the product $(x+6)(x+5)$ and its expansion is $x^2 + 11x + 30$ .

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Question Expand the expression (x+6)(x+5)(x+6)(x+5)(x+6)(x+5) and illustrate the resulting product as a rectangle.

Studdy Solution

STEP 1

STEP 2

To illustrate the product of (x+6)(x+6)(x+6) and (x+5)(x+5)(x+5) as a rectangle, we will draw a rectangle and divide it into parts that represent each term in the product.

STEP 3

The first term of each binomial, xxx, will represent one dimension of the rectangle. The second terms, 666 and 555, will represent the other dimension.

STEP 4

Draw a rectangle and label one side with x+6x+6x+6 and the adjacent side with x+5x+5x+5.

STEP 5

Divide the rectangle into four smaller parts by drawing a line parallel to the side labeled x+5x+5x+5 at a distance of 666 units from it and another line parallel to the side labeled x+6x+6x+6 at a distance of 555 units from it.

STEP 6

Label the four smaller rectangles/squares that result from the division. The top left square will be x×xx \times xx×x, the top right rectangle will be x×5x \times 5x×5, the bottom left rectangle will be 6×x6 \times x6×x, and the bottom right rectangle will be 6×56 \times 56×5.

STEP 7

STEP 8

The completed rectangle now illustrates the product (x+6)(x+5)(x+6)(x+5)(x+6)(x+5) with each part of the rectangle representing a term in the expanded form of the product.

STEP 9

The expanded form of the product can be found by adding the areas of all the smaller rectangles/squares.

STEP 10

Write down the expanded form using the areas of the smaller parts:x2+5x+6x+30x^2 + 5x + 6x + 30x2+5x+6x+30

STEP 11

Combine like terms in the expanded form to simplify the expression.x2+(5x+6x)+30x^2 + (5x + 6x) + 30x2+(5x+6x)+30

STEP 12

Calculate the sum of the like terms 5x5x5x and 6x6x6x.5x+6x=11x5x + 6x = 11x5x+6x=11x

STEP 13

Write the final simplified form of the expanded product.x2+11x+30x^2 + 11x + 30x2+11x+30The rectangle illustrates the product (x+6)(x+5)(x+6)(x+5)(x+6)(x+5) and its expansion is x2+11x+30x^2 + 11x + 30x2+11x+30.

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Question Expand the expression $(x+6)(x+5)$ and illustrate the resulting product as a rectangle.

To illustrate the product of $(x+6)$ and $(x+5)$ as a rectangle, we will draw a rectangle and divide it into parts that represent each term in the product.

The first term of each binomial, $x$ , will represent one dimension of the rectangle. The second terms, $6$ and $5$ , will represent the other dimension.

Draw a rectangle and label one side with $x+6$ and the adjacent side with $x+5$ .

Divide the rectangle into four smaller parts by drawing a line parallel to the side labeled $x+5$ at a distance of $6$ units from it and another line parallel to the side labeled $x+6$ at a distance of $5$ units from it.

Label the four smaller rectangles/squares that result from the division. The top left square will be $x \times x$ , the top right rectangle will be $x \times 5$ , the bottom left rectangle will be $6 \times x$ , and the bottom right rectangle will be $6 \times 5$ .

The completed rectangle now illustrates the product $(x+6)(x+5)$ with each part of the rectangle representing a term in the expanded form of the product.

Write down the expanded form using the areas of the smaller parts:
$x^2 + 5x + 6x + 30$

Combine like terms in the expanded form to simplify the expression.
$x^2 + (5x + 6x) + 30$

Calculate the sum of the like terms $5x$ and $6x$ .
$5x + 6x = 11x$

Write the final simplified form of the expanded product.
$x^2 + 11x + 30$
The rectangle illustrates the product $(x+6)(x+5)$ and its expansion is $x^2 + 11x + 30$ .