Simplify 5⁄2 ÷ 1⁄x
Answer: A
Dividing by a fraction means multiplying by its reciprocal.
\[ \frac{5}{2} \div \frac{1}{x} = \frac{5}{2} \times \frac{x}{1} = \frac{5x}{2} \]
The reciprocal of \( \tfrac{1}{x} \) is \( x \), so the \( x \) moves to the numerator.
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Simplify a(c - b) - b(a - c)
Answer: B
Expand both brackets, watching the sign in front of the second.
\[ a(c-b) - b(a-c) = ac - ab - ab + bc \]
\[ = ac - 2ab + bc \]
The minus outside the second bracket flips both of its terms, which is what turns \( -bc \) into \( +bc \) and produces the \( -2ab \).
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Coefficient of x² in 4x³ + 3x² - x + 1 is:
Answer: C
Read off the number multiplying \( x^{2} \) in
\[ 4x^{3} + 3x^{2} - x + 1 \]
The coefficient is 3. The \( -1 \) belongs to \( x \) and the \( 4 \) to \( x^{3} \).
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Expand and simplfy (x - 5)(x + 4)
Answer: D
\[ (x-5)(x+4) = x^{2} + 4x - 5x - 20 \]
\[ = x^{2} - x - 20 \]
The constants \( -5 \) and \( +4 \) sum to \( -1 \) for the middle term and multiply to \( -20 \) for the last.
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Expand and simplfy (x - y)(x + y)
Answer: C
\[ (x-y)(x+y) = x^{2} + xy - xy - y^{2} = x^{2} - y^{2} \]
The two middle terms cancel, which is exactly the difference-of-squares identity. The options with \( \pm 2xy \) are the expansions of \( (x \mp y)^{2} \).
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(a - b)² =
Answer: A
\[ (a-b)^{2} = (a-b)(a-b) = a^{2} - ab - ab + b^{2} \]
\[ = a^{2} - 2ab + b^{2} \]
The middle term is essential: \( (a-b)^{2} \) is not \( a^{2} - b^{2} \), which factorises as \( (a+b)(a-b) \) instead.
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Factorise x² + x - 72
Answer: C
Find two numbers whose product is \( -72 \) and whose sum is \( +1 \).
\[ 9 \times (-8) = -72, \qquad 9 + (-8) = 1 \]
\[ x^{2} + x - 72 = (x - 8)(x + 9) \]
Checking: \( x^{2} + 9x - 8x - 72 = x^{2} + x - 72 \). Swapping the signs would give a middle term of \( -x \).
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Factorise -20x² - 9x + 20
Answer: A
Test the candidate factorisation by expanding it.
\[ (5 + 4x)(4 - 5x) = 20 - 25x + 16x - 20x^{2} \]
\[ = -20x^{2} - 9x + 20 \]
which matches. The constant \( +20 \) needs both signs the same in the products \( 5 \times 4 \), and the \( -20x^{2} \) needs the two \( x \)-terms to have opposite signs - only this arrangement does both.
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Expand and simplify (x + y)³
Answer: B
\[ (x+y)^{3} = x^{3} + 3x^{2}y + 3xy^{2} + y^{3} \]
The two middle terms share a factor of \( 3xy \):
\[ 3x^{2}y + 3xy^{2} = 3xy(x + y) \]
\[ \therefore\ (x+y)^{3} = x^{3} + 3xy(x+y) + y^{3} \]
The sign inside the bracket must match the original: \( (x-y)^{3} = x^{3} - 3xy(x-y) - y^{3} \).
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Simplify (x - 9)(x + 10) ⁄ (x² - 81)
Answer: C
Factorise the denominator as a difference of squares, then cancel.
\[ x^{2} - 81 = (x-9)(x+9) \]
\[ \frac{(x-9)(x+10)}{(x-9)(x+9)} = \frac{x+10}{x+9} \]
The common factor \( (x-9) \) cancels, provided \( x \neq 9 \). Cancelling the wrong pair would leave \( \tfrac{x+10}{x-9} \).
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