Algebra Questions and Answers

Q: On evaluating expression 10(x - 20y) + 100x, answer will be

\[ 10(x - 20y) = 10x - 200y \]\[ 10x - 200y + 100x = 110x - 200y \]The 10 multiplies both terms inside the bracket, so the \( 20y \) becomes \( 200y \).

Q: Expand and simplfy (x - 5)(x + 4)

\[ (x-5)(x+4) = x^{2} + 4x - 5x - 20 \]\[ = x^{2} - x - 20 \]The constants \( -5 \) and \( +4 \) sum to \( -1 \) for the middle term and multiply to \( -20 \) for the last.

Q: If algebraic expression is 2(x - y) + 3(4x -5y) then answer will be

Expand and collect.\[ 2(x - y) = 2x - 2y \]\[ 3(4x - 5y) = 12x - 15y \]\[ (2x + 12x) + (-2y - 15y) = 14x - 17y \]Both \( y \) terms are negative, so their coefficients add in magnitude.

Q: Expand and simplfy (x - y)(x + y)

\[ (x-y)(x+y) = x^{2} + xy - xy - y^{2} = x^{2} - y^{2} \]The two middle terms cancel, which is exactly the difference-of-squares identity. The options with \( \pm 2xy \) are the expansions of \( (x \mp y)^{2} \).

Q: On evaluating algebraic expression -3(4s - 2t) - 4(2s + 3t), answer will be

Expand both brackets, distributing the negative signs.\[ -3(4s - 2t) = -12s + 6t \]\[ -4(2s + 3t) = -8s - 12t \]\[ (-12s - 8s) + (6t - 12t) = -20s - 6t \]The \( -3 \) turns \( -2t \) into \( +6t \), which is the sign most often lost.

Q: Factorise -20x² - 9x + 20

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.

Q: a²-b² is a product of

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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141.

By factorizing algebraic expression 36ab - 48bc, answer must be

A.6b(6a + 8c)
B.8b(2a - 5c)
C.6b(6a - 8c)
D.7b(5a - 9c)

Answer: C

Look for the common factors of the coefficients and the variables.

\[ 36ab - 48bc = 6b(6a - 8c) \]

Checking: \( 6b \times 6a = 36ab \) and \( 6b \times 8c = 48bc \).

The variable \( b \) appears in both terms and comes out; \( a \) and \( c \) do not. A fuller factorisation is \( 12b(3a - 4c) \), but among the options only \( 6b(6a - 8c) \) expands back correctly.

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142.

On evaluating expression 10(x - 20y) + 100x, answer will be

A.100x + 240y
B.110x + 200y
C.200x - 110y
D.110x - 200y

Answer: D

\[ 10(x - 20y) = 10x - 200y \]

\[ 10x - 200y + 100x = 110x - 200y \]

The 10 multiplies both terms inside the bracket, so the \( 20y \) becomes \( 200y \).

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143.

Answer of factorization of expression (3x - 2y)(a + b) + (4x - 3y)(a + b) is

A.(5x + 5y)(a + b)
B.(5x + 5y) - (a + b)
C.(6x - 8y) + (a - b)
D.(7x - 5y)(a + b)

Answer: D

Both terms share the factor \( (a + b) \), so take it out first:

\[ (3x - 2y)(a+b) + (4x - 3y)(a+b) = \left[(3x - 2y) + (4x - 3y)\right](a+b) \]

Collecting like terms inside the bracket:

\[ 3x + 4x = 7x, \qquad -2y - 3y = -5y \]

\[ \therefore\; (7x - 5y)(a + b) \]

The sign is the trap: both \( y \) terms are negative, so they add to \( -5y \) rather than cancelling.

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144.

Expand and simplfy (x - 5)(x + 4)

A.x² + 9x - 20
B.x² - x - 9
C.x² - x - 1
D.x² - x - 20

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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145.

If algebraic expression is 2(x - y) + 3(4x -5y) then answer will be

A.14x - 17y
B.17x - 14y
C.12x - 18y
D.18x + 30y

Answer: A

Expand and collect.

\[ 2(x - y) = 2x - 2y \]

\[ 3(4x - 5y) = 12x - 15y \]

\[ (2x + 12x) + (-2y - 15y) = 14x - 17y \]

Both \( y \) terms are negative, so their coefficients add in magnitude.

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146.

Expand and simplfy (x - y)(x + y)

A.x² - 2xy + y²
B.x² + 2xy + y²
C.x² - y²
D.x² + y²

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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147.

On evaluating algebraic expression -3(4s - 2t) - 4(2s + 3t), answer will be

A.−20s - 6t
B.−6s + 20t
C.21s - 7t
D.23s + 8t

Answer: A

Expand both brackets, distributing the negative signs.

\[ -3(4s - 2t) = -12s + 6t \]

\[ -4(2s + 3t) = -8s - 12t \]

\[ (-12s - 8s) + (6t - 12t) = -20s - 6t \]

The \( -3 \) turns \( -2t \) into \( +6t \), which is the sign most often lost.

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148.

Factorise -20x² - 9x + 20

A.(5 + 4x)(4 - 5x)
B.(5 - 4x)(4 - 5x)
C.(5 - 4x)(4 + 5x)
D.(5 + 4x)(4 + 5x)

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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149.

pqr is what type of polynomial?

A.Monomial
B.Binomial
C.Trimonial
D.None of these

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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150.

a²-b² is a product of

A.(a+b)(a-b)
B.(a+b)(a+b)
C.(a-b)(a-b)
D.None of these

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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