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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By factorizing expression 2bx + 4by - 3ax -6ay, answer must be
Answer: B
Group the terms in pairs and factor each.
\[ 2bx + 4by - 3ax - 6ay = 2b(x + 2y) - 3a(x + 2y) \]
The bracket \( (x + 2y) \) is now common:
\[ = (2b - 3a)(x + 2y) \]
Taking out \( -3a \) rather than \( 3a \) from the second pair is what makes both brackets identical.
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Algebraic expression of sum of 10a and 15c can be written as
Answer: A
"Sum" means addition, and the two terms are unlike - one in \( a \), the other in \( c \) - so they cannot be combined.
\[ 10a + 15c \]
Only like terms may be added; \( 25ac \) would be the result of a quite different operation.
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Algebraic expression of word expression "subtract eight times of number b from ten times of number c" is
Answer: D
"Subtract eight times \( b \) from ten times \( c \)" means starting with \( 10c \) and taking \( 8b \) away.
\[ 10c - 8b \]
The phrase "subtract A from B" always puts B first - writing \( 8b - 10c \) reverses the two and gives the negative of the answer.
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If a = -12 and b = -14 then 2a + 2b + 3 is
Answer: B
Substitute \( a = -12 \) and \( b = -14 \).
\[ 2a = -24, \qquad 2b = -28 \]
\[ -24 - 28 + 3 = -49 \]
Both doubled values stay negative; the \( +3 \) only softens the total slightly.
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Which of the following is the numerical coefficient of x²y²?
Answer: A
The numerical coefficient is the number multiplying the variable part. Written in full, the term is
\[ x^{2}y^{2} = 1 \cdot x^{2}y^{2} \]
so the coefficient is 1. \( x^{2} \) and \( y^{2} \) are parts of the variable factor, not numbers.
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If the expression \( 4c + 4b + 4d \) is subtracted from \( 3c + 4b + 5d \), then the answer will be
Answer: A
Subtracting flips every sign in the expression being taken away.
\[ (3c + 4b + 5d) - (4c + 4b + 4d) \]
\[ = (3c - 4c) + (4b - 4b) + (5d - 4d) \]
\[ = -c + d \]
The \( b \) terms cancel exactly, which is why no \( b \) survives.
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If 6x + 2y - 4z is subtracted from 2x - 8y + 5z then answer will be
Answer: D
Subtracting flips every sign in the second expression.
\[ (2x - 8y + 5z) - (6x + 2y - 4z) \]
\[ = 2x - 8y + 5z - 6x - 2y + 4z \]
\[ = -4x - 10y + 9z \]
The \( -4z \) becomes \( +4z \), which is what makes the \( z \) coefficient \( +9 \).
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The value of x² - 5 at x= -1 is -
Answer: C
Substitute \( x = -1 \), squaring before subtracting.
\[ x^{2} - 5 = (-1)^{2} - 5 = 1 - 5 = -4 \]
A negative number squared is positive, so \( (-1)^{2} = 1 \), not \( -1 \) - that slip would give \( -6 \).
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In 15y and 160ab, coefficients are
Answer: D
The coefficient is the numerical factor of a term.
\[ 15y \rightarrow 15, \qquad 160ab \rightarrow 160 \]
The letters \( y \), \( a \) and \( b \) are the variable parts, not coefficients.
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