Chapter 10 · Exercise 2Textbook pages 183–184

Exercise 2, Questions 1–2 – Solutions ബഹുപദമാണോ?

Six situations. For each, write the relation as an algebraic expression, then decide: is it a polynomial? The reason is what earns the marks – so always say what is done to x.

📖 What you need (page 8 of this guide)

Question 1 (i) · The path around a square ground

Q1. ചുവടെയുള്ള കണക്കുകളിലെല്ലാം, പറഞ്ഞിരിക്കുന്ന അളവുകൾ തമ്മിലുള്ള ബന്ധം ബഹുപദമാണോ എന്നു പരിശോധിക്കുക. In each situation, check whether the relation between the quantities is a polynomial, and give the reason.

(i) A square ground has a path 1 metre wide all around it. The relation between the length of a side of the ground and the area of the path.

Call the side x m. The ground with the path is a bigger square of side x + 2. Path area = big square − ground. Or cut the path into strips and corners.

🎬 Solution animation – cutting up the path
🧪 Try any ground
Answer: with side x m, path area = (x + 2)² − x² = 4x + 4 sq m. This is a polynomial (first degree): x is only multiplied by the fixed number 4, and the fixed number 4 is added. (If the path were inside the ground, the area would be x² − (x − 2)² = 4x − 4 – also a polynomial.)

Question 1 (ii) · Adding acid to a liquid

(ii) A liquid is made of 7 litres of water and 3 litres of acid. More acid is poured in. The relation between the amount of acid added and the change in the percentage of acid in the liquid.

Before: 3 out of 10 litres is acid. After adding x litres: 3 + x out of 10 + x litres. Percentage = part ÷ whole × 100. Subtract the old 30%.

🎬 Solution animation – building the expression
🧪 Pour in acid

However much acid you pour, the percentage never reaches 100%, so the change never reaches 70. The change grows more and more slowly – a polynomial like 4x + 4 could not behave like this.

x (litres added)041030
acid %, 100(3 + x)/(10 + x)30506582.5
change, 70x/(10 + x)0203552.5
Answer: change in percentage = 100(3 + x)/(10 + x) − 30 = 70x/(10 + x). This is not a polynomial: we divide by 10 + x, an expression containing the changing number x.

Question 1 (iii) · The rope between two poles

(iii) Two poles 3 m and 4 m high stand 5 m apart. A rope is pulled from the top of one pole, fixed to the ground, and then pulled to the top of the second pole. The relation between the distance from the foot of a pole to the point where the rope is fixed, and the length of the rope.

The rope makes two right triangles with the ground. Use Pythagoras for each part.

🎬 Solution animation – two right triangles
🖐️ Move the point where the rope is fixed

Can you find where the rope is shortest? (Near x ≈ 2.14 m, length √74 ≈ 8.60 m.)

Answer: with x m from the foot of the 3 m pole, rope length = √(x² + 9) + √((5 − x)² + 16) m. Not a polynomial: it takes square roots of expressions that contain the changing number x.
💡 Why this matters

Real measurements often lead to roots and divisions. Recognising when a relation is (or is not) a polynomial tells us which tools we can use – polynomials are the easiest kind of rule to calculate with, graph and study.

Question 2 · Three operations

Q2. ചുവടെപ്പറഞ്ഞിരിക്കുന്ന ക്രിയകളോരോന്നും ബീജഗണിതവാചകമായി എഴുതുക. Write each operation as an algebraic expression. Explain which of them are polynomials.

(i) The sum of a number and its reciprocal. (ii) The sum of a number and its square root. (iii) The product of (the number plus its square root) and (the number minus its square root).

For (iii), use (a + b)(a − b) = a² − b² and remember (√x)² = x.

🎬 Solution animation
🧪 Test with a number
Answer: (i) x + 1/x – not a polynomial (reciprocal of x). (ii) x + √x – not a polynomial (square root of x). (iii) (x + √x)(x − √x) = x² − (√x)² = x² − x – a polynomial of degree 2: the square roots cancel out! (for x ≥ 0, where √x exists)
💡 Why this matters

Judge an expression by what it is after simplifying, not by how it first looks. (x + √x)(x − √x) looks full of roots, but it is just x² − x.

Check yourself

Next: Finding p(0), p(1), p(−1), p(10) ▶