Algebraic identities

Correct answer: B. \((x+y)^2=x^2+2xy+y^2\)

Option B follows the square-of-a-sum identity: \((x+y)^2=x^2+2xy+y^2\), so it is true for every x and y. In A, the \(2xy\) term is missing. Exam tip: expand both sides to test an identity.

Correct answer: A. \(a^2-b^2=(a+b)(a-b)\)

Since \(49m^2=(7m)^2\) and \(25n^2=(5n)^2\), the expression is a difference of two squares. Therefore, use \(a^2-b^2=(a+b)(a-b)\). Exam tip: first check whether both terms are perfect squares.

Correct answer: B. (-40a)

The expansion is (16a^2-40a+25), so like terms cancel leaving (-40a). Exam tip: do not forget the negative middle term.

Correct answer: A. \(a^2-b^2=(a+b)(a-b)\)

The difference-of-squares identity is \(a^2-b^2=(a+b)(a-b)\). On multiplying, the middle terms \(ab\) and \(-ab\) cancel. \((a-b)^2\) contains an extra \(-2ab\) term. Exam tip: check the signs before choosing an identity.

Correct answer: A. 2436

Here, \(58=50+8\) and \(42=50-8\). Using the identity \((a+b)(a-b)=a^2-b^2\), we get \((50+8)(50-8)=50^2-8^2=2500-64=2436\). Option 2500 is only \(50^2\); subtracting \(8^2\) is essential. Exam tip: when two numbers are equally distant from a common middle number, use the difference of squares identity.

Correct answer: C. (a+b)

A coefficient is the number or algebraic expression that multiplies a particular variable term. First expand the product using the distributive property: \\( (x+a)(x+b)=x\cdot x+x\cdot b+a\cdot x+a\cdot b\\). This gives \\(x^2+bx+ax+ab\\). Combining the two terms containing x gives \\(x^2+(a+b)x+ab\\). Therefore, the coefficient of x is \\(a+b\\), so option C is correct.

It is important to identify the term, not just any expression in the expansion. The coefficient of \\(x^2\\) is 1, and the constant term is \\(ab\\), but neither is the coefficient of x. The two contributions to the x-term are \\(bx\\) and \\(ax\\); adding them produces \\((a+b)x\\). This reasoning works for arbitrary values of a and b, including zero or negative values. Hence the third choice gives the required coefficient.

Correct answer: B. (-22)

The direct answer is B: the constant term is −22. A constant term is the part with no variable, so in (x−2)(x+11) it is obtained by multiplying the two constant numbers, −2 and 11. Step by step, expand: x·x=x², x·11=11x, (−2)·x=−2x, and (−2)·11=−22. Combining the middle terms gives x²+9x−22; therefore the term without x is −22. Option A, 22, has the wrong sign because a negative number multiplied by a positive number is negative. Option B is correct because (−2)×11=−22. Option C, 9, is the coefficient of x after combining 11x and −2x, not the constant term. Option D, −9, is not produced by the expansion and confuses the middle-term calculation with the constant calculation. Exam cue: in a product of two linear expressions, multiply the constants to find the constant term, while checking their signs carefully. Decision: pass; B is mathematically correct.

Correct answer: A. \(4x^2-8x-21\)

The direct answer is option A, \(4x^2-8x-21\). Expand by multiplying every term in the first bracket by every term in the second. First, \((2x)(2x)=4x^2\). Next, \((2x)(-7)=-14x\), and \(3(2x)=6x\). Finally, \(3(-7)=-21\). Combine like terms: \(-14x+6x=-8x\), so the result is \(4x^2-8x-21\). Option A contains all three correct parts. Option B has the wrong sign for the linear term; the two linear products give a negative total. Option C uses \(-14x\) but forgets the additional \(+6x\), and it also gives the wrong constant sign. Option D has the wrong leading coefficient because \(2x\times2x=4x^2\), not \(2x^2\). Exam cue: use FOIL or distribute carefully, then combine only like terms.

Correct answer: B. \(2ab\)

\((a+b)^2=a^2+2ab+b^2\). Subtracting \(a^2+b^2\) gives \(2ab\). \(ab\) is only the product and misses the coefficient 2 that arises in the binomial expansion. Exam tip: whenever expanding \((a+b)^2\), check that the middle term is \(2ab\).

Correct answer: A. (2b^2-2ab)

( (a-b)^2=a^2-2ab+b^2 ), so subtracting (a^2-b^2) gives (2b^2-2ab). Exam tip: keep both identities separate.

Correct answer: A. 53

Using the identity \((x+y)^2=x^2+y^2+2xy\), we get \(x^2+y^2=(x+y)^2-2xy\). Therefore, \(x^2+y^2=9^2-2\times14=81-28=53\). Option 81 is only the value of \((x+y)^2\); subtracting \(2xy\) is necessary. Exam tip: when \(x+y\) and \(xy\) are given, apply this identity directly to find the sum of squares.

Correct answer: B. 37

Using the identity \((p-q)^2=p^2-2pq+q^2\), we get \(p^2+q^2=(p-q)^2+2pq\). Therefore, \(p^2+q^2=5^2+2\times6=25+12=37\). Option 25 is only the value of \((p-q)^2\); the term \(2pq\) must also be added. Exam tip: when \(p-q\) and \(pq\) are given, apply \(p^2+q^2=(p-q)^2+2pq\) directly.

Correct answer: A. \(a^2-b^2=(a+b)(a-b)\)

The difference of two squares, \(a^2-b^2\), factorises as \((a+b)(a-b)\). Options B and C are square-expansion identities, while D should contain \(-b^2\), not \(+b^2\). Exam tip: for a difference of squares, look for sum and difference factors.

Correct answer: A. \((x^2+3x+2)(x+3)\)

Multiply the first two factors: \((x+1)(x+2)=x^2+2x+x+2=x^2+3x+2\). Hence the correct intermediate step is \((x^2+3x+2)(x+3)\). In option B, the linear terms have been combined incorrectly. Exam tip: when multiplying two binomials, multiply every term in one bracket by every term in the other bracket.

Correct answer: A. \(4x+8\)

The common factor in both terms is \(x+2\): \((x+2)^2-(x+2)(x-2)=(x+2)[(x+2)-(x-2)]\). The bracket equals \(4\), so the expression becomes \(4(x+2)=4x+8\). \(4x\) is incomplete because it misses the constant term \(8\). Exam tip: In expressions involving subtraction, factor out a common factor before expanding whenever possible.

Correct answer: A. \(18x^2+8\)

Using the identity \((a+b)^2+(a-b)^2=2a^2+2b^2\), put \(a=3x\) and \(b=2\): \(2(3x)^2+2(2)^2=18x^2+8\). Therefore, \(18x^2+8\) is correct. \(9x^2+4\) misses the doubling that occurs when both squared expressions are added. Exam tip: in such a pair, the mixed terms with opposite signs cancel out.

Correct answer: B. 4

Use the identity \((a-b)^2=a^2-2ab+b^2\). Here, \(a=2x\) and \(b=5\), so \((2x-5)^2=4x^2-20x+25\). Therefore, the coefficient of \(x^2\) is 4. Note that 20 is associated with the middle term, \(-20x\), not with \(x^2\). Exam tip: always check the power of the variable before identifying a coefficient.

Correct answer: C. 9

Use the identity \((a+b)^2=a^2+2ab+b^2\). Here, \(a=x\) and \(b=3y\), so \((x+3y)^2=x^2+6xy+9y^2\). Therefore, the coefficient of \(y^2\) is 9. The number 6 is the coefficient of the middle term \(6xy\), not of \(y^2\). Exam tip: always square the numerical coefficient in the last term.

Correct answer: A. The first is (x^2-25) and the second is (x^2-10x+25)

In opposite-sign product mixed terms cancel but in a square the middle term remains. Exam tip: do not mix the two forms.

Correct answer: A. \(a^2+2ab+b^2\)

\((a+b)^2=(a+b)(a+b)=a^2+2ab+b^2\), so A is correct. In B, the sign of the last term is incorrect. Exam tip: the middle term is always twice the product of the two terms.

Correct answer: B. (-2ab)

The direct answer is option B: \(-2ab\). Start with the identity \\(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx\\), taking the three terms as \(a\), \(-b\), and \(c\). The product involving \(a\) and \(-b\) is \(2(a)(-b)=-2ab\). Therefore the required term is negative. Option A, \(2ab\), would result from \(a+b\), not \(a-b\). Option B is correct because the negative sign belonging to \(-b\) remains in the mixed term. Option C, \(ab\), misses the factor 2 required in a square expansion. Option D, \(-ab\), also misses that factor 2. Remember: in a square, every pair of unlike terms gives twice their product; include the signs before multiplying.

Correct answer: A. 0

The governing concept is the pair of square identities: (x+y)^2=x^2+2xy+y^2 and (x-y)^2=x^2-2xy+y^2. Subtract the second expansion from the first. The x^2 and y^2 terms cancel, while 2xy-(-2xy)=4xy. The complete expression is therefore 4xy-4xy=0. Hence option A is correct. A common error is to stop after evaluating only the difference of the first two squared expressions; that gives 4xy, which is option B, but the original expression still contains the final subtraction of 4xy. Option C incorrectly treats the result as half of the intermediate term, and option D is not produced by subtracting these two square expansions. The cancellation must be completed before choosing the answer.

Correct answer: B. \(8xy\)

Use the identity \((a+b)^2-(a-b)^2=4ab\). Here, \(a=2x\) and \(b=y\), so the expression becomes \(4\times 2x\times y=8xy\). \(4xy\) is a close distractor, but it misses the factor 2 in \(a=2x\). Exam tip: recognise the \(4ab\) identity directly in such expressions.

Correct answer: A. \((a-b)^2=a^2-2ab+b^2\)

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.

Correct answer: A. \(x^2+14x+49\)

In \(x^2+14x+49\), the last term is \(49=7^2\) and the middle term is \(14x=2\cdot x\cdot7\). Hence it is \((x+7)^2\). In exams, always check whether the middle term equals \(2ab\).