What are the factors of (12a^2-7a-10)?
Correct answer: D
Expanding ((4a-5)(3a+2)) gives (12a^2+8a-15a-10). Exam tip: add opposite-signed terms carefully.
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Factorisation
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Hard · Level 70 · quadratic factorisation,hard algebra,signs
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- ((3a+2)(4a-5))
- ((3a-5)(4a+2))
- ((2a-5)(6a+2))
- ((4a-5)(3a+2))
Hard · Level 70 · factorisation, algebraic identities, difference of squares, perfect square trinomials, class 9 mathematics
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- \(49a^2-b^2\)
- \(49a^2+b^2\)
- \(49a^2-14ab+b^2\)
- \(49a^2+14ab+b^2\)
Hard · Level 70 · factorisation, algebraic identities, perfect square trinomial, class 9 mathematics
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- \(x^2+10x+25\)
- \(x^2+10x+20\)
- \(x^2-25\)
- \(x^2+5x+25\)
Hard · Level 70 · perfect square,binomial square,factorisation
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- ((2m-3n)^2)
- ((4m+9n)^2)
- ((2m+3n)^2)
- ((m+3n)(4m+3n))
Hard · Level 70 · trinomial factorisation,signs,quadratic
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- ((x-6)(x-7))
- ((x+6)(x+7))
- ((x-3)(x-14))
- ((x+3)(x-14))
Hard · Level 70 · factorisation, algebraic identities, difference of cubes, class 9 mathematics, polynomials
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- \(x^3-8y^3\)
- \(x^3+8y^3\)
- \(x^2-4y^2\)
- \(x^3-4y^3\)
Hard · Level 70 · perfect square,minus identity,factorisation
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- ((4a-5b)^2)
- ((4a+5b)^2)
- ((16a-25b)^2)
- ((8a-5b)^2)
Hard · Level 70 · algebraic identities, factorisation, perfect square trinomial, class 9 mathematics, quadratic expressions
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- \(9x^2-24xy+16y^2\)
- \(9x^2+24xy-16y^2\)
- \(9x^2-16y^2\)
- \(9x^2-24xy+8y^2\)
Hard · Level 70 · algebraic identities, factorisation, perfect square trinomial, class 9 mathematics, binomial square
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- \(16p^2-24pq+9q^2\)
- \(16p^2-24pq+6q^2\)
- \(16p^2-9q^2\)
- \(16p^2+24pq-9q^2\)
Hard · Level 70 · quadratic factorisation,checking,algebra
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- ((3y+2)(y+4))
- ((3y+4)(y+2))
- ((y+2)(3y+8))
- ((3y+8)(y+1))
Hard · Level 70 · cube identity,factorisation,algebraic identities
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- ((3p-2q)(9p^2+6pq+4q^2))
- ((3p+2q)(9p^2-6pq+4q^2))
- ((27p-8q)(p^2+pq+q^2))
- ((3p-2q)(9p^2-6pq+4q^2))
Hard · Level 70 · sum of cubes,factorisation,identity
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- ((x+4)(x^2-4x+16))
- ((x-4)(x^2+4x+16))
- ((x+4)(x^2+4x+16))
- ((x+8)(x^2-8x+64))
Hard · Level 70 · common factor,three terms,factorisation
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- (5xy(x-3y+2))
- (5x(xy-3y^2+2y))
- (xy(5x-15y+10))
- (10xy(x-3y+2))
Hard · Level 70 · factorisation, algebraic identities, perfect square trinomial, class 9 mathematics, polynomial classification
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- मध्य पद, पहले और अंतिम पद के वर्गमूलों के गुणनफल का दोगुना होना चाहिए।
- मध्य पद, पहले और अंतिम पदों के योग के बराबर होना चाहिए।
- मध्य पद, पहले और अंतिम पदों के गुणनफल के बराबर होना चाहिए।
- मध्य पद का गुणांक हमेशा 1 होना चाहिए।
Hard · Level 70 · complete factorisation,difference of squares,higher powers
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- ((x^2-9)(x^2+9))
- ((x-3)(x+3)(x^2+9))
- ((x-9)(x+9))
- ((x-3)^2(x+3)^2)
Hard · Level 70 · factorisation by grouping,algebraic identities,common binomial,class 9 mathematics,polynomial factorisation
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- \((3a+5)(2b+3)\)
- \((3a+5)(2b+5)\)
- \((3a+10)(2b+3)\)
- \((6a+5)(b+3)\)
Hard · Level 70 · trinomial,two variables,factorisation
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- ((x+3y)(x-2y))
- ((x-3y)(x+2y))
- ((x+6y)(x-y))
- ((x-6y)(x+y))
Hard · Level 70 · algebraic identities, factorisation, perfect square trinomial, quadratic trinomials, class 9 mathematics
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- \(a^2-14ab+49b^2\)
- \(a^2-14ab+14b^2\)
- \(a^2+14ab+49b^2\)
- \(a^2-7ab+49b^2\)
Hard · Level 70 · two variables,hard factorisation,signs
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- ((3x-2y)(4x-5y))
- ((3x-5y)(4x-2y))
- ((6x-5y)(2x-2y))
- ((12x-10y)(x-y))
Hard · Level 70 · sum of cubes,identity,factorisation
View options
- ((a+b)(a^2-ab+b^2))
- ((a-b)(a^2+ab+b^2))
- ((a+b)(a^2+ab+b^2))
- ((a-b)(a^2-ab+b^2))
Which of the following expressions can be factorised using the identity of difference of squares, \(a^2-b^2=(a-b)(a+b)\)?
Correct answer: A
\(49a^2-b^2=(7a)^2-b^2\), so it is the difference of two perfect squares. Applying \(a^2-b^2=(a-b)(a+b)\) gives \((7a-b)(7a+b)\). Option B is a sum of squares, while C and D are \((7a-b)^2\) and \((7a+b)^2\), respectively. Exam tip: for a difference of squares, check that there is a minus sign between the two square terms.
View question detailsWhich of the following expressions is a perfect-square trinomial that can be written as the product of two identical binomial factors?
Correct answer: A
In \(x^2+10x+25\), the end terms are \(x^2\) and \(5^2\), and the middle term is \(2\cdot x\cdot5\). Hence it equals \((x+5)^2\). In \(x^2+10x+20\), 20 is not \(5^2\). Exam tip: always verify the middle term using \(2ab\).
View question detailsWhat is the correct factorised form of (4m^2+12mn+9n^2)?
Correct answer: C
Here (4m^2=(2m)^2), (9n^2=(3n)^2), and the middle term is (2(2m)(3n)). Exam tip: match all three terms.
View question detailsIn which form will (x^2-13x+42) be written?
Correct answer: A
Since (-6-7=-13) and ((-6)(-7)=42). Exam tip: use two negative signs for positive last term and negative middle term.
View question detailsWhich of the following expressions is a difference of two cubes and can be factorised using the relevant identity?
Correct answer: A
Since \(8y^3=(2y)^3\), \(x^3-8y^3=x^3-(2y)^3\) is a difference of cubes. Use \(a^3-b^3=(a-b)(a^2+ab+b^2)\). \(x^3+8y^3\) is a sum of cubes. Exam tip: first rewrite the coefficient as a perfect cube.
View question detailsWhose perfect square is (16a^2-40ab+25b^2)?
Correct answer: A
Here (16a^2=(4a)^2) and (25b^2=(5b)^2), with middle term (-2(4a)(5b)). Exam tip: identify negative perfect squares.
View question detailsWhich of the following expressions can be factorised using the identity \((a-b)^2\)?
Correct answer: A
Here, \(9x^2=(3x)^2\), \(16y^2=(4y)^2\), and \(-24xy=-2(3x)(4y)\). Thus A is \((3x-4y)^2\). In B, the last term is negative. Exam tip: always verify the middle term.
View question detailsWhich of the following expressions is a perfect-square trinomial and therefore has two identical binomial factors?
Correct answer: A
Here, \(16p^2=(4p)^2\), \(9q^2=(3q)^2\), and \(-24pq=-2(4p)(3q)\). Hence it is \((4p-3q)^2\). Exam tip: match the middle term with \(\pm2ab\).
View question detailsFactorise (3y^2+14y+8).
Correct answer: B
((3y+2)(y+4)) gives (12y+2y), while ((3y+4)(y+2)) is correct. Exam tip: verify by expansion.
View question detailsWhich is the correct factorised form for (27p^3-8q^3)?
Correct answer: A
This is ((3p)^3-(2q)^3), and (a^3-b^3=(a-b)(a^2+ab+b^2)). Exam tip: identify cube roots first.
View question detailsWhat is the correct factorised form of (x^3+64)?
Correct answer: A
Since (64=4^3), use the (a^3+b^3) formula. Exam tip: keep the middle sign negative in sum of cubes.
View question detailsWhat is the simplest factorised form of (5x^2y-15xy^2+10xy)?
Correct answer: A
A common factor is a factor that appears in every term. Here the three terms are products of numbers and variables, so we look for the greatest factor shared by all of them. The factor must be taken outside the bracket, leaving a simpler expression inside.
Each term contains 5, x, and y. Taking out \(5xy\), we get \(5xy(x-3y+2)\), because \(5xy\cdot x=5x^2y\), \(5xy\cdot(-3y)=-15xy^2\), and \(5xy\cdot2=10xy\). Thus option A is the simplest factorised form. Options B and C are equivalent but do not take out the greatest common factor completely, while D gives twice the original expression.
If the first and last terms of a trinomial are perfect squares, what condition must its middle term satisfy for the trinomial to be identified as a perfect-square trinomial?
Correct answer: A
A perfect-square trinomial has the form \(a^2 \pm 2ab + b^2\). Hence its middle term must be \(\pm 2ab\), twice the product of the square roots of the outer terms. Perfect-square outer terms alone are not enough. Exam tip: check the sign of the middle term too.
View question detailsWhat is the complete factorised form of (x^4-81)?
Correct answer: B
First (x^4-81=(x^2-9)(x^2+9)), then (x^2-9=(x-3)(x+3)). Exam tip: factor further when possible.
View question detailsFactorise (6ab+9a+10b+15) by grouping.
Correct answer: A
Group the terms as \(6ab+9a+10b+15=3a(2b+3)+5(2b+3)\). The common binomial is \((2b+3)\), so the factorisation is \((2b+3)(3a+5)\), which is equivalent to option A. In option B, using \((2b+5)\) does not produce the correct middle and constant terms. Exam tip: after grouping, look for an identical common factor in both groups.
View question detailsWhat are the correct factors of (x^2+xy-6y^2)?
Correct answer: A
The sum of (3y) and (-2y) is (y), and the product is (-6y^2). Exam tip: watch the signs of (y)-terms.
View question detailsWhich of the following quadratic trinomials can be written as the product of two identical binomials?
Correct answer: A
In \(a^2-14ab+49b^2\), the first and last terms are \(a^2\) and \((7b)^2\). Its middle term is \(-2\times a\times7b=-14ab\), so it equals \((a-7b)^2\). Exam tip: check whether the middle term is \(\pm2pq\).
View question detailsWhat are the factors of (12x^2-23xy+10y^2)?
Correct answer: A
((3x-2y)(4x-5y)) gives (12x^2-15xy-8xy+10y^2). Exam tip: add both negative middle terms.
View question detailsWhich is the correct factorised form of (a^3+b^3)?
Correct answer: A
For sum of cubes, the first factor is (a+b) and the second is (a^2-ab+b^2). Exam tip: remember the order of signs.
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