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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Which of the following is the numerical coefficient of -5xy?
Answer: C
\[ -5xy = (-5) \cdot xy \]
The numerical coefficient is −5 - the sign belongs with it. Dropping the minus gives 5, which is the magnitude of the coefficient rather than the coefficient itself.
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pqr is what type of polynomial?
Answer: A
Polynomials are named by their number of terms, and terms are separated by \( + \) or \( - \).
\( pqr \) is a single product with no addition or subtraction, so it is one term - a monomial.
Three variables do not make three terms; \( p + q + r \) would be a trinomial.
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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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a²-b² is a product of
Answer: A
This is the difference of two squares.
\[ a^{2} - b^{2} = (a+b)(a-b) \]
Expanding checks it: \( a^{2} - ab + ab - b^{2} = a^{2} - b^{2} \), the middle terms cancelling. \( (a+b)^{2} \) and \( (a-b)^{2} \) both leave a \( \pm 2ab \) term behind.
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Which of the following is the value of (x+ 1/x)²?
Answer: C
Apply \( (p+q)^{2} = p^{2} + 2pq + q^{2} \) with \( p = x \) and \( q = \tfrac{1}{x} \).
\[ \left(x + \frac{1}{x}\right)^{2} = x^{2} + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^{2}} \]
\[ = x^{2} + \frac{1}{x^{2}} + 2 \]
The middle term simplifies to the constant 2 because \( x \) cancels with \( \tfrac{1}{x} \) - which is why this identity is so useful for problems linking \( x + \tfrac{1}{x} \) to \( x^{2} + \tfrac{1}{x^{2}} \).
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Which of the following is obtained by subtracting x²-y² from y² - x² ?
Answer: A
Subtracting means taking the first expression away from the second.
\[ (y^{2} - x^{2}) - (x^{2} - y^{2}) = y^{2} - x^{2} - x^{2} + y^{2} \]
\[ = 2y^{2} - 2x^{2} = -2(x^{2} - y^{2}) \]
The bracket's signs both flip when it is subtracted - missing that gives 0 instead.
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What degree does x³ - x²y² - 8y²+ 2 have?
Answer: C
The degree of a polynomial is the highest total degree among its terms.
The largest is 4. In a term with several variables the exponents are added, which is what puts \( x^{2}y^{2} \) above \( x^{3} \).
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What is the product of (x+a) and (x+b)?
Answer: D
Expand term by term.
\[ (x+a)(x+b) = x^{2} + bx + ax + ab \]
\[ = x^{2} + (a+b)x + ab \]
The coefficient of \( x \) is the sum of the two constants and the last term their product - the standard pattern used when factorising quadratics.
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Simplify 15ax² ⁄ 5x
Answer: B
Divide the coefficients and subtract the exponents.
\[ \frac{15ax^{2}}{5x} = \frac{15}{5} \cdot a \cdot x^{2-1} = 3ax \]
The \( x \) in the denominator lowers the power by one; leaving it as \( x^{2} \) is the usual slip.
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