Algebraic identities

Multiply the first two factors: \((x+1)(x+2)=x^2+2x+x+2=x^2+3x+2\). Hence, the correct intermediate form is \((x^2+3x+2)(x+3)\). In option B, the linear terms have been combined incorrectly, while option D has the wrong sign for the constant term. Exam tip: while multiplying two binomials, add the middle terms separately before simplifying.

On expanding, \((2x-3)(2x+9)=4x^2+18x-6x-27=4x^2+12x-27\). Therefore, the coefficient of \(x\) is \(12\). The value \(18\) comes from only one cross-product, \(2x\times9\); the term \((-3)\times2x=-6x\) must also be included. Exam tip: Combine all linear terms before identifying the coefficient of \(x\).

The cross terms are (11x) and (-5x), so the total is (6x). Exam tip: take the algebraic sum of numbers for the coefficient of (x).

Use the identity \\(a+b)^2-(a-b)^2=4ab\\). Here, the first square has the form \\(a+b)^2\\) and the second has the form \\(a-b)^2\\), with \\(a=2x\\) and \\(b=5\\). Therefore the difference is \\(4(2x)(5)=40x\\). The squared terms and the constant squared terms cancel when the expressions are expanded.

Hence option B, \\(40x\\), is correct. Direct expansion gives \\(4x^2+20x+25-(4x^2-20x+25)\\); cancelling like terms leaves \\(40x\\). Option A, \\(20x\\), misses one of the two middle terms, while option D contains terms that cancel. The identity is the quickest reliable method.

In ( (a-b)^2 ) the middle term is negative. Exam tip: keep both coefficient squares and (2ab) correct.

The product of two binomials can be expanded by multiplying every term in the first bracket by every term in the second bracket. In \\((x+6)(x-2)\\), the products are \\(x\cdot x=x^2\\), \\(x\cdot(-2)=-2x\\), \\(6\cdot x=6x\\), and \\(6\cdot(-2)=-12\\). Combining the middle terms gives \\(-2x+6x=4x\\).

Therefore, the complete expansion is \\(x^2+4x-12\\), so option B is correct. A useful check is the pattern \\((x+r)(x+s)=x^2+(r+s)x+rs\\). Here, \\(r+s=6-2=4\\) and \\(rs=6(-2)=-12\\). Option A incorrectly adds 6 and 2, ignoring the negative sign.

The expression is a difference of two squares, so use \\(a^2-b^2=(a+b)(a-b)\\). Taking \\(a=104\\) and \\(b=96\\), we get \\(104^2-96^2=(104+96)(104-96)=200\\times8=1600\\). This is faster and safer than calculating both large squares separately.

Therefore, option C is correct. The key is to preserve the subtraction and use the sum and difference of the two numbers. The other values do not result from this product: \\(800\\) is only the difference multiplied by 100, \\(1200\\) is too small, and \\(2000\\) incorrectly uses the sum without the correct difference. The identity directly confirms 1600.

The expansion of \((a+b)^2\) contains the middle term \(+2ab\), twice the product of \(a\) and \(b\). In \((a-b)^2\), the middle term is \(-2ab\). Exam tip: the sign between the terms determines the sign of the middle term.

In the square of three terms, twice the product of each pair appears. Exam tip: identify the pair for mixed terms.

(2\times2x\times3=12x) gives the linear (x) term. Exam tip: identify the mixed term formed with the constant separately.

\(x^2-25=x^2-5^2\), so it is a difference of two squares and factorises as \((x-5)(x+5)\). Option D is a perfect-square trinomial. Exam tip: check for two squared terms separated by a minus sign.

The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\), so option A is correct. Option B gives \((x-y)^2=x^2-2xy+y^2\), not \(x^2-y^2\). Exam tip: use conjugate factors for a difference of squares.

(87\times93=(90-3)(90+3)=90^2-3^2=8091). Exam tip: use (a^2-b^2) for numbers around the same centre.

( (4x-3)(4x+3)=16x^2-9 ), and (+9) gives (16x^2). Exam tip: identify difference of squares first.

To find both requested values, first expand the product or use the pattern for two binomials. The constant term comes from multiplying the two constants, 2 and 7, giving 14. The coefficient of x comes from the two middle products: x multiplied by 7 and 2 multiplied by x. These give 7x and 2x, whose sum is 9x. Therefore, the constant term and coefficient of x are 14 and 9 respectively, so option A is correct.

In full form, \\( (x+2)(x+7)=x^2+7x+2x+14=x^2+9x+14 \\). The number at the end is the constant term, while the number attached to x is its coefficient. Option B reverses these two values, and options C and D use individual numbers rather than the required product and sum. Keeping the sum for the middle coefficient and the product for the constant avoids this common error.

(121=11^2) and the middle term is (-22x=-2\times x\times11). Exam tip: identify ((a-b)^2) from the negative middle term.

Use the identity \((x+y)^2-(x-y)^2=4xy\). Here, \(x=7a\) and \(y=2b\), so the value is \(4\times 7a\times 2b=56ab\). \(49a^2-4b^2\) is the value of \((7a+2b)(7a-2b)\), not the difference of the two squares given here. Exam tip: first identify the pattern \((x+y)^2-(x-y)^2\).

Using the identity \((a+b)^2+(a-b)^2=2a^2+2b^2\), take \(a=2p\) and \(b=3q\). This gives \(2(2p)^2+2(3q)^2=8p^2+18q^2\). The mixed terms involving \(12pq\) cancel when the two squares are added, so option C is not correct. Exam tip: recognise this identity first to avoid lengthy expansion.

The expansion of \((a-b)^2\) is \(a^2-2ab+b^2\), so option A is correct. Option B represents the square of a sum. Exam tip: always check the sign of the middle term.

(197=200-3), so ( (200-3)^2=40000-1200+9=38809 ). Exam tip: subtract the middle term in ( (a-b)^2 ).