Algebraic identities

Option A is correct because multiplying \((a-b)(a-b)\) gives \(a^2-ab-ab+b^2=a^2-2ab+b^2\). Option B is the expansion of \((a+b)^2\). Exam tip: check the sign of the middle term carefully.

Using ( (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) ), (36=x^2+y^2+z^2+22). Exam tip: apply the three-term identity directly.

This is ( (2x)^3+3^3 ), so the sum of cubes identity applies. Exam tip: identify cube roots first.

Using \((a+b)^2+(a-b)^2=2a^2+2b^2\), take \(a=2x\) and \(b=3y\). This gives \(2(2x)^2+2(3y)^2=8x^2+18y^2\). The \(24xy\) terms cancel when the two squares are added, so option C is not correct. Exam tip: recognise the \((a+b)^2+(a-b)^2\) pattern and apply the identity directly.

Apply the identity \((x+y)^2-(x-y)^2=4xy\). Here, \(x=4a\) and \(y=b\), so the expression becomes \(4\times4a\times b=16ab\). \(32ab\) results from introducing an extra factor of 2, so it is incorrect. Exam tip: Recognise this difference-of-squares identity before expanding the brackets.

Use the identity \((a-b)^2=a^2-2ab+b^2\). Taking \(a=3x\) and \(b=5\), we get \((3x-5)^2=9x^2-30x+25\). In option A, the middle term would be \(+30x\), so it is not correct. Exam tip: take the square roots of the first and last terms, then verify the middle term using \(\pm2ab\).

Here, \(9x^2=(3x)^2\) and \(16y^2=(4y)^2\). The middle term is \(2\times3x\times4y=24xy\), so the expression is \((3x+4y)^2\). Exam tip: always verify the middle term.

\(121a^2-49b^2=(11a)^2-(7b)^2\). Applying the difference-of-squares identity \(x^2-y^2=(x+y)(x-y)\) gives \((11a+7b)(11a-7b)\). Options B and C are perfect squares, so their expansions contain a middle term \(\pm154ab\), which is absent in the given expression. Exam tip: first rewrite each term as a perfect square, then check for the difference-of-squares identity.

Expanding the expression,


a(3x-4)(3x+7)=9x^2+21x-12x-28=9x^2+9x-28.

Therefore, the coefficient of (x) is 9. Here, 21 is the coefficient from only one cross-term; the other cross-term, −12x, must also be combined. Exam tip: After multiplying two binomials, always combine like terms.

First expand the product: \((x+4)(x-9)=x^2-9x+4x-36=x^2-5x-36\). Subtracting \(x^2\) gives \(x^2-x^2-5x-36=-5x-36\). Therefore, option B is correct. In option A, the sign of the \(x\)-term is incorrect. Exam tip: while multiplying binomials, check the signs of the middle terms carefully.

Since \((U+V)^2=U^2+2UV+V^2\) and \((U-V)^2=U^2-2UV+V^2\), \(U^2+V^2\) remains unchanged. Only the middle term changes sign. Exam tip: compare the outer terms first.

A has end terms \((4p)^2\), \((3q)^2\) and middle term \(-2(4p)(3q)=-24pq\), so it is \((4p-3q)^2\). B has a negative final term. Tip: check the middle-term sign.

Using (a^2-b^2=(a+b)(a-b)), (2000\cdot16=32000). Exam tip: do not calculate both squares separately.

Write \(997=1000-3\). Using \((a-b)^2=a^2-2ab+b^2\), \((1000-3)^2=1000^2-2\times1000\times3+3^2=1000000-6000+9=994009\). Therefore, option A is correct. \(9940090\) has an extra zero. Exam tip: for squares of numbers close to 1000, use \((a-b)^2\).

\(a^2-b^2=(a+b)(a-b)\) is a difference of squares, so it is not generally the square of a binomial. In contrast, A, B and D are \((a+b)^2\), \((a-b)^2\) and \((2a+b)^2\). Exam tip: use the middle term to identify the binomial signs.

In \((x+y+z)^2\), the square terms appear along with twice each pairwise product. Hence A is correct. In B, the mixed-term coefficients are 1, not 2. Exam tip: check every mixed-term coefficient.

Use the identity \\( (u-v)^2-(u+v)^2=-4uv\\). Here, u=5m and v=2n. The first square has the minus form and the second square has the plus form, so their difference is negative. Substituting gives \\(-4(5m)(2n)=-40mn\\). Therefore option B is correct. The order of subtraction matters; changing it would change the sign of the result.

This can also be checked by expansion. The first square is \\(25m^2-20mn+4n^2\\), and the second is \\(25m^2+20mn+4n^2\\). Subtracting the second from the first cancels the two square terms and gives \\(-20mn-20mn=-40mn\\). Thus neither 40mn nor 20mn has the correct sign or magnitude. The expressions 25m² and 4n² are only parts of the expansion, not the final simplified difference.

The correct identity is \(a^2-b^2=(a-b)(a+b)\). On multiplying the factors, the middle terms cancel, leaving \(a^2-b^2\). Option B is incorrect because \((a-b)^2\) contains the middle term \(-2ab\). Exam tip: look for paired factors with opposite signs.

Here \(9a^2=(3a)^2\) and \(16b^2=(4b)^2\), while the middle term is \(-2\times3a\times4b=-24ab\). So it is \((3a-4b)^2\). In A, the middle term should be \(8x\), not \(10x\). Exam tip: verify the middle term using \(\pm2ab\).

Multiply the first two factors: \((x+6)(x-4)=x^2-4x+6x-24=x^2+2x-24\). Therefore, option A is correct. In option B, the middle term is incorrect, while option D has the wrong sign for the constant term. Exam tip: when multiplying two binomials, check the sum of the inner and outer terms for the middle term.