Visual models of identities

The area is ((x+1)^2), and the expansion is (x^2+2x+1). Exam tip: remember to add the two (x)-rectangles.

The small square has side (2), so its area is (2^2=4). Exam tip: connect the constant term with the small square.

In the area model, the rectangle is split into four parts with areas \(x\cdot x=x^2\), \(x\cdot1=x\), \(3\cdot x=3x\), and \(3\cdot1=3\). Thus, the total area is \(x^2+x+3x+3=x^2+4x+3\). Option D has constant term 4, but the constant part here must be \(3\times1=3\). Exam tip: multiply every term of one binomial by every term of the other, then combine like terms.

When the square is divided into lengths a and b, two rectangles are formed, each with area ab. Thus, they contribute the middle term 2ab. The areas a² and b² belong to the corner squares. Exam tip: add the two equal rectangle areas carefully.

Dividing a square of side \(m+n\) according to lengths \(m\) and \(n\) gives four parts: \(m^2\), \(mn\), \(mn\), and \(n^2\). Hence, the term \(mn\) appears twice. \(2mn\) is the sum of the two \(mn\) rectangles, not the area of one part. Exam tip: use the area model to remember \((m+n)^2=m^2+2mn+n^2\).

The whole area of a square is the square of its side. With side \\(p+5\\), the area is \\((p+5)^2\\). Applying \\((a+b)^2=a^2+2ab+b^2\\), with \\(a=p\\) and \\(b=5\\), gives \\(p^2+2(p)(5)+5^2=p^2+10p+25\\). The middle term represents the two equal rectangles in the visual model.

Therefore, option D is correct. The model contains one square of area \\(p^2\\), two rectangles whose total area is \\(5p+5p=10p\\), and one small square of area \\(25\\). Option A misses one of the two rectangles, while options B and C place the coefficients with the wrong terms. Adding all parts gives the complete area, so D follows both visually and algebraically.

The area of a square is the square of its side, so the area is \((a-b)^2\). Using the identity \((x-y)^2=x^2-2xy+y^2\), with \(x=a\) and \(y=b\), gives \(a^2-2ab+b^2\). The expression \(a^2-b^2\) is the expansion of \((a-b)(a+b)\), not of \((a-b)^2\). Exam tip: In \((a-b)^2\), the middle term is always \(-2ab\).

The total area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\), so the model represents \((a+b)^2\). In \((a-b)^2\), the middle term is negative. Exam tip: count both \(ab\) rectangles.

In (a^2-b^2), the smaller (b^2) square is removed from the larger (a^2) square. Exam tip: identify the removed square.

Area of a rectangle = length × breadth = \((r+s)(r-s)\). Using the identity \((a+b)(a-b)=a^2-b^2\), the area is \(r^2-s^2\). Option B is the expansion of \((r-s)^2\), so it does not apply here. Exam tip: when a sum and difference are multiplied, the middle terms cancel.

The total area of the two rectangles is (2\cdot10\cdot1=20). Exam tip: link the middle part with two equal rectangles.

The side of the square is \(20-1=19\). Therefore, its area is \((20-1)^2=19^2\), so the model represents the square of 19. Choosing 20 is incorrect because it is the number before subtraction. Exam tip: Evaluate the expression inside the brackets first, then identify the square.

Dividing each side \(a+b\) into lengths \(a\) and \(b\) forms areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Hence, there are two \(ab\) rectangles. Exam tip: count the two middle regions to identify \(2ab\).

Dividing a square of side a+b into lengths a and b gives one a² square, one b² square, and two ab rectangles. The two rectangles together give 2ab. In exams, count both mixed rectangles, not just one.

In the rectangle model, the constant part is the small rectangle with side lengths 2 and 4. Its area is \(2\times4=8\). The value 6 comes from \(2+4\), which is related to the coefficient of the middle term \(6x\), not to the constant area. Exam tip: to find the constant term, multiply only the numerical terms.

Since \(x^2+2xy+y^2=(x+y)^2\), it is the area of a square of side \(x+y\). The model has one \(x^2\) tile, two \(xy\) tiles, and one \(y^2\) tile. Exam tip: look for the middle term \(2xy\).

In the square model, the larger component square has side \(2x\). Therefore, its area is \((2x)^2=4x^2\). In \(2x^2\), the coefficient 2 has not been squared, while \(2x\) is only the side length, not the area. Exam tip: To find the area of a square, square its side length.

Add the areas of all regions: \(a^2+ab+ab+b^2=a^2+2ab+b^2\). This is the area model of \((a+b)^2\). Option A subtracts the two \(ab\) regions. Exam tip: count the rectangular parts to identify the coefficient of the middle term.

In the area model for \((a+b)^2\), the large square is divided into regions of area \(a^2\), \(b^2\), and two equal rectangles of area \(ab\) each. The two rectangles together give \(ab+ab=2ab\). In \((a-b)^2\), the middle term is \(-2ab\), so it is not correct. Exam tip: always check the sign of the middle term in a squared binomial.

The large square has side (10), so its area is (10^2=100). Exam tip: identify large and small squares separately.