Exploring Algebraic Identities — Practice

Question 1

Expand (x + 5)².

Accepted Answer

Step 1 — (a+b)² with a = x, b = 5.

Step 2 — x² + 2(x)(5) + 25 = x² + 10x + 25.

✦ Answer: x² + 10x + 25.

Question 2

Factorise x² − 49.

Accepted Answer

Step 1 — a² − b² with a = x, b = 7.

Step 2 — (x + 7)(x − 7).

✦ Answer: (x + 7)(x − 7).

Question 3

Use an identity to find 102 × 98.

Accepted Answer

Step 1 — (100 + 2)(100 − 2) = 100² − 2².

Step 2 — 10000 − 4 = 9996.

✦ Answer: 9996.

Question 4

Expand (3a − 2b)².

Accepted Answer

Step 1 — (a−b)² with a = 3a, b = 2b.

Step 2 — (3a)² − 2(3a)(2b) + (2b)² = 9a² − 12ab + 4b².

✦ Answer: 9a² − 12ab + 4b².

Question 5

If a + b = 10 and ab = 21, find a² + b².

Accepted Answer

Step 1 — a² + b² = (a+b)² − 2ab.

Step 2 — = 10² − 2(21) = 100 − 42 = 58.

✦ Answer: 58.

Question 6

Factorise 27x³ + 1.

Accepted Answer

Step 1 — a³ + b³ with a = 3x, b = 1.

Step 2 — (3x + 1)((3x)² − (3x)(1) + 1²) = (3x + 1)(9x² − 3x + 1).

✦ Answer: (3x + 1)(9x² − 3x + 1).

Question 7

Expand (x + 2y + 3z)².

Accepted Answer

Step 1 — (a+b+c)² = a²+b²+c²+2ab+2bc+2ca with a=x, b=2y, c=3z.

Step 2 — x² + 4y² + 9z² + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x).

Step 3 — x² + 4y² + 9z² + 4xy + 12yz + 6zx.

✦ Answer: x² + 4y² + 9z² + 4xy + 12yz + 6zx.

Question 8

Evaluate 8³ + (−5)³ + (−3)³ using an identity.

Accepted Answer

Step 1 — here a + b + c = 8 + (−5) + (−3) = 0.

Step 2 — when a + b + c = 0, a³ + b³ + c³ = 3abc.

Step 3 — 3abc = 3 × 8 × (−5) × (−3) = 360.

✦ Answer: 360.

Practice — Exploring Algebraic Identities