Visual models of identities

Both rectangles are (6b) and (6b), so the total is (12b). In exams, add strips from both sides.

The visual model represents a large square divided into a square part, two rectangular strips, and a small corner square. The first square has area \\(c^2\\), so its side is \\(c\\). Each strip has area \\(9c\\), which agrees with a length of \\(c\\) multiplied by a width of \\(9\\). The corner has area \\(81=9^2\\), so its side is \\(9\\).

The complete side is therefore the side of the first square plus the side of the corner: \\(c+9\\). Equivalently, the model gives \\(c^2+18c+81=(c+9)^2\\). Hence option A is correct. Option B incorrectly adds the strip widths twice as a single side, and option C is not a length obtained from the square model.

(4x^2=(2x)^2), (49=7^2) and the middle term should be (2\cdot2x\cdot7=28x), so the given total strips do not match. For the correct square, total strips must be (28x).

A square model forms two equal rectangles, so (6x) appears. In exams, do not make the one-strip mistake.

The remaining area is \(x^2-y^2\). By the difference-of-squares identity, \(x^2-y^2=(x+y)(x-y)\), so A is correct. \((x-y)^2\) is the area of a smaller square, not the leftover region. Exam tip: a removed square usually signals difference of squares.

While removing two strips, the (b^2) corner is removed twice. In exams, add it back once.

In the rectangle model, one side is split as \(x+2\) and the other as \(x+7\). The four small rectangle areas are \(x\times x=x^2\), \(x\times 7=7x\), \(2\times x=2x\), and \(2\times 7=14\). Therefore, \(x^2, 2x, 7x, 14\) is the correct set. Option C incorrectly uses \(9\) instead of the constant product \(14\). Exam tip: find each part by multiplying the two side lengths of its small rectangle.

In the area model, the cross-products give the linear terms \(2x\times4=8x\) and \(1\times x=x\). Therefore, the total \(x\)-term is \(8x+x=9x\). \(8x\) is only one cross-product, not the complete linear term. Exam tip: combine only like terms with the same power of \(x\).

The area of the rectangle is the product of its sides: \((3y+2)(y+5)\). The four parts of the rectangle have areas \(3y^2\), \(15y\), \(2y\), and \(10\). Therefore, the total area is \(3y^2+15y+2y+10=3y^2+17y+10\). Option B misses the \(2y\) term. Exam tip: while using the distributive property, multiply every term in one bracket by every term in the other bracket.

In a grid model, the constant corner is the product of the constant terms of the two binomials. Here, the constant terms are 3 and 2, so the constant corner is \(3\times2=6\). Options 3 and 2 are individual constants, while 5 is their sum, not their product. Exam tip: identify the variable and constant terms separately before filling the four grid regions.

The total model area is \(a^2+ab+ab+b^2=a^2+2ab+b^2\), which equals \((a+b)^2\). In \((a-b)^2\), the middle term is \(-2ab\). Exam tip: identify the two \(ab\) rectangles.

Both strips are (6x) and (6x), so total (12x) is subtracted. In exams, count both removed parts.

The area of a square is the square of its side, so the expression is \((2x-5)^2\). Using \((a-b)^2=a^2-2ab+b^2\), with \(a=2x\) and \(b=5\), gives \(4x^2-20x+25\). In option D, the sign of the middle term is wrong: the middle term is negative for the square of a difference. Exam tip: while finding \(-2ab\), multiply by the complete term \(2x\).

On expansion, (x^2-16x+64) is subtracted from (x^2+16x+64), leaving (32x). In exams, cancel like terms.

Because of three pairs of rectangles, (2ab), (2bc) and (2ca) appear. In exams, count each pair twice.

A square whose side is \\(a+b+c\\) can be divided into three smaller squares and several rectangles. The three square regions have side lengths \\(a\\), \\(b\\), and \\(c\\). The area of a square is its side multiplied by itself, so their areas are \\(a^2\\), \\(b^2\\), and \\(c^2\\).

Therefore, option A gives the correct set of small-square areas. Products such as \\(ab\\), \\(bc\\), and \\(ca\\) describe rectangular regions, because their two side lengths are different. The expressions \\(2ab\\), \\(2bc\\), and \\(2ca\\) describe combined areas of pairs of rectangles in an identity model, not the individual small squares. Thus the distinction between square regions and rectangular regions leads to A.

In both expansions, (m^2) and (n^2) cancel, leaving (2mn-(-2mn)=4mn). In exams, treat minus of minus as addition.

Both models have the (x) by (x) part (x^2). In exams, identify the common large square first.

Each side of the square is \(x+5\), so its area is \((x+5)^2=x^2+2\times x\times5+25=x^2+10x+25\). Hence, the two equal rectangular strips together give \(10x\), the middle term. The constant term is \(25\), not \(10\). Exam tip: in \((a+b)^2\), the middle term is always \(2ab\).

This is a direct visual model of difference of squares. In exams, match ((a+b)(a-b)) with (a^2-b^2).