Question 16 of 43intermediate🔧 ApplyShort Answer2 marks

Lay out algebra tiles for x2x^{2} + 11x + 30 in such a way that you will see its factors.

Correct Answer

To lay out the algebra tiles for x2+11x+30x^2 + 11x + 30, we must split the middle term 11x11x as 5x+6x5x + 6x. We start by placing one x2x^2-tile in the top-left corner. Next, we place five xx-tiles on the right side of the x2x^2-tile and six xx-tiles arranged below it.

Then, we fill the remaining space by arranging the 30 unit tiles in a 5 by 6 array. Once this rectangular arrangement is formed, we observe that the dimensions of the rectangle are x+5x + 5 and x+6x + 6 units respectively. This visually reveals that the factors of x2+11x+30x^2 + 11x + 30 are (x+5)(x + 5) and (x+6)(x + 6).

Exercise: Think and Reflect | Q: 2 | (Page 12)
For More Understanding

Explanation

The textbook explains that to factor using algebra tiles, the x-term must be split such that the product of the split coefficients equals the constant term, and their sum equals the coefficient of x. For x2+11x+30x^2 + 11x + 30, splitting 11x as 5x + 6x allows the 30 unit tiles to be arranged perfectly in a 5 by 6 array, forming a complete rectangle with dimensions representing the factors.

Solution Steps

  1. Step 1: Split the middle term 11x11x as 5x+6x5x + 6x (since 5+6=115 + 6 = 11 and 5×6=305 \times 6 = 30).

  2. Step 2: Place one x2x^2-tile, five xx-tiles to its right, and six xx-tiles below it.

  3. Step 3: Arrange the 30 unit tiles in a 5 by 6 array to complete the rectangle.

  4. Step 4: Read the dimensions of the completed rectangle as (x+5)(x + 5) and (x+6)(x + 6) to identify the factors.

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