Algebraic Expressions – Step-by-Step Solutions
a) 2y(y + 4)
Step 1: Multiply 2y with each term inside the bracket.
2y × y = 2y²
2y × 4 = 8y
Answer: 2y² + 8y
b) (y + 5)(y + 2)
Use the distributive property (FOIL method).
y × y = y²
y × 2 = 2y
5 × y = 5y
5 × 2 = 10
Now combine like terms: 2y + 5y = 7y
Answer: y² + 7y + 10
c) (2 − t)(1 − 2t)
Use distributive property:
2 × 1 = 2
2 × (−2t) = −4t
(−t) × 1 = −t
(−t) × (−2t) = 2t²
Now combine: 2 − 4t − t + 2t² = 2t² − 5t + 2
Answer: 2t² − 5t + 2
d) (x − 4)(x + 4)
This is a difference of squares formula:
(x − a)(x + a) = x² − a²
So, x² − 16
Answer: x² − 16
e) −(4 − x)(x + 4)
Rewriting (4 − x) as (−x + 4)
Apply distributive property:
−[(4 × x) + (4 × 4) − (x × x) − (x × 4)]
= −(4x + 16 − x² − 4x)
Simplify: −(−x² + 16)
= x² − 16
Answer: x² − 16
f) −(a + b)(b − a)
This is a product of sum and difference: (a + b)(b − a)
= b² − a²
Now apply the negative sign: −(b² − a²) = a² − b²
Answer: a² − b²
f) −(a + b)(b − a)
Step 1: Use identity (a + b)(b − a) = b² − a²
= −(b² − a²)
= a² − b²
Answer: a² − b²
See Also: Exercise 1-8
g) (2p + 9)(3p + 1)
Step 1: Apply distributive property (FOIL)
2p × 3p = 6p²
2p × 1 = 2p
9 × 3p = 27p
9 × 1 = 9
Now combine like terms: 2p + 27p = 29p
Answer: 6p² + 29p + 9
h) (3k − 2)(k + 6)
Step 1: Apply distributive property
3k × k = 3k²
3k × 6 = 18k
−2 × k = −2k
−2 × 6 = −12
Combine like terms: 18k − 2k = 16k
Answer: 3k² + 16k − 12
i) (s + 6)²
Step 1: Use identity (a + b)² = a² + 2ab + b²
= s² + 2 × s × 6 + 6²
= s² + 12s + 36
Answer: s² + 12s + 36
j) −(7 − x)(7 + x)
Step 1: Use identity (a − b)(a + b) = a² − b²
= −(49 − x²)
= x² − 49
Answer: x² − 49
k) (3x − 1)(3x + 1)
Step 1: Use identity (a − b)(a + b) = a² − b²
= (3x)² − 1²
= 9x² − 1
Answer: 9x² − 1
l) (7k + 2)(3 − 2k)
Step 1: Apply distributive property
7k × 3 = 21k
7k × (−2k) = −14k²
2 × 3 = 6
2 × (−2k) = −4k
Combine like terms: 21k − 4k = 17k
Answer: −14k² + 17k + 6
m) (1 − 4x)²
Step 1: Use identity (a − b)² = a² − 2ab + b²
a = 1, b = 4x
= 1² − 2 × 1 × 4x + (4x)²
= 1 − 8x + 16x²
Answer: 16x² − 8x + 1
n) (−3 − y)(5 − y)
Step 1: Apply distributive property
−3 × 5 = −15
−3 × (−y) = 3y
−y × 5 = −5y
−y × (−y) = y²
Combine terms: 3y − 5y = −2y
Answer: y² − 2y − 15
o) (8 − x)(8 + x)
Step 1: Use identity (a − b)(a + b) = a² − b²
= 64 − x²
Answer: 64 − x²
p) (9 + x)²
Step 1: Use identity (a + b)² = a² + 2ab + b²
a = 9, b = x
= 81 + 18x + x²
Answer: x² + 18x + 81
q) (−7y + 11)(−12y + 3)
Step 1: Apply distributive property
−7y × (−12y) = 84y²
−7y × 3 = −21y
11 × (−12y) = −132y
11 × 3 = 33
Combine like terms: −21y − 132y = −153y
Answer: 84y² − 153y + 33
r) (g − 5)²
Step 1: Use identity (a − b)² = a² − 2ab + b²
= g² − 10g + 25
Answer: g² − 10g + 25
s) (d + 9)²
Step 1: Use identity (a + b)² = a² + 2ab + b²
= d² + 2 × d × 9 + 81
= d² + 18d + 81
Answer: d² + 18d + 81
t) (6d + 7)(6d − 7)
Step 1: Use identity (a + b)(a − b) = a² − b²
a = 6d, b = 7
= (6d)² − 7² = 36d² − 49
Answer: 36d² − 49
u) (5z + 1)(5z − 1)
Step 1: Use identity (a + b)(a − b) = a² − b²
= (5z)² − 1² = 25z² − 1
Answer: 25z² − 1
v) (1 − 3h)(1 + 3h)
Step 1: Use identity (a − b)(a + b) = a² − b²
a = 1, b = 3h
= 1 − 9h²
Answer: 1 − 9h²
w) (2p + 3)(2p + 2)
Step 1: Use distributive property
2p × 2p = 4p²
2p × 2 = 4p
3 × 2p = 6p
3 × 2 = 6
Combine like terms: 4p + 6p = 10p
Answer: 4p² + 10p + 6
x) (8a + 4)(a + 7)
Step 1: Use distributive property
8a × a = 8a²
8a × 7 = 56a
4 × a = 4a
4 × 7 = 28
Combine like terms: 56a + 4a = 60a
Answer: 8a² + 60a + 28
y) (5r + 4)(2r + 4)
Step 1: Use distributive property
5r × 2r = 10r²
5r × 4 = 20r
4 × 2r = 8r
4 × 4 = 16
Combine like terms: 20r + 8r = 28r
Answer: 10r² + 28r + 16
z) (w + 1)(w − 1)
Step 1: Use identity (a + b)(a − b) = a² − b²
= w² − 1
Answer: w² − 1
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