Chapter Test Paper — Set 1

(b) 3x² − 5x + 2

Reason:
(a) x + 1/x = x + x⁻¹ — has a negative exponent, so not a polynomial.
(b) 3x² − 5x + 2 — all exponents are whole numbers. This is a polynomial.
(c) √x + 1 = x^1/2 + 1 — exponent 1/2 is not a whole number, so not a polynomial.
(d) 2^x — the variable is in the exponent, so not a polynomial.

3.
The highest power of x in 4x³ − 7x + 9 is 3 (from the term 4x³), so the degree is 3.

Binomial.
A polynomial with exactly two terms is called a binomial. Example: x + 5, 3x² − 1.

False.
The degree of the zero polynomial is not defined. It is not 0. (The constant polynomial p(x) = c, where c ≠ 0, has degree 0. But the zero polynomial p(x) = 0 has no defined degree.)

p(x) = 5x − 10 = 0
5x = 10
x = 10/5 = 2.
The zero of p(x) = 5x − 10 is x = 2.

p(x) = 7 is a constant polynomial with degree 0.
(It has no variable term; the highest power of x is x⁰ = 1, so degree = 0.)

(i) 3x² → Monomial (one term)
(ii) x + 5 → Binomial (two terms)
(iii) x² − x + 1 → Trinomial (three terms)
(iv) −4x³ → Monomial (one term)
[½ mark each]

(i) 5x³ − 2x² + x − 7: The term with x² is −2x², so coefficient of x² = −2. [1 mark]

(ii) 3 − x + x²: The term with x² is x² (i.e., 1·x²), so coefficient of x² = 1. [1 mark]

Finding the zero:
p(x) = 3x + 9 = 0
3x = −9
x = −9/3 = −3 [1 mark]

Verification:
p(−3) = 3(−3) + 9 = −9 + 9 = 0 ✓ [1 mark]
Since p(−3) = 0, x = −3 is indeed the zero of p(x).

No, 1/x + 3 is not a polynomial. [1 mark]

Reason: 1/x = x⁻¹. In a polynomial, all exponents of the variable must be non-negative integers (whole numbers). Since −1 is a negative exponent, the expression 1/x + 3 does not satisfy the definition of a polynomial. [1 mark]

General form: A linear polynomial is of the form ax + b, where a ≠ 0 and a, b are real numbers. [1 mark]

Examples:
(i) 3x + 7  (here a = 3, b = 7)
(ii) −2x + 5  (here a = −2, b = 5) [1 mark]

(i) p(x) = 4x − 8:
4x − 8 = 0 ⇒ 4x = 8 ⇒ x = 2
Verify: p(2) = 4(2) − 8 = 8 − 8 = 0 ✓ [1 mark]

(ii) p(x) = −3x + 15:
−3x + 15 = 0 ⇒ −3x = −15 ⇒ x = 5
Verify: p(5) = −3(5) + 15 = −15 + 15 = 0 ✓ [1 mark]

(iii) p(x) = ½x − 3:
½x − 3 = 0 ⇒ ½x = 3 ⇒ x = 6
Verify: p(6) = ½(6) − 3 = 3 − 3 = 0 ✓ [1 mark]

Polynomial	By Degree	By No. of Terms
(i) x³ + 2x + 1	Cubic (degree 3)	Trinomial (3 terms)
(ii) 5	Constant (degree 0)	Monomial (1 term)
(iii) −2x²	Quadratic (degree 2)	Monomial (1 term)
(iv) x² + x − 3	Quadratic (degree 2)	Trinomial (3 terms)

[½ mark for each correct degree, ½ mark for each correct classification by terms; total = ½ × 4 + ½ × 4 = can get partial. Award 3 marks for all correct.]

(i) x² + √2·x + 1:
Yes, it is a polynomial. The coefficient √2 is an irrational number, but all exponents of x (2, 1, 0) are whole numbers. Coefficients can be any real number. [¾ mark]

(ii) x + 1/x:
Not a polynomial. 1/x = x⁻¹ has a negative exponent. Exponents must be non-negative integers. [¾ mark]

(iii) √x + 3:
Not a polynomial. √x = x^1/2; the exponent 1/2 is not a whole number. [¾ mark]

(iv) 2x³ − 5x + 7:
Yes, it is a polynomial. All exponents of x (3, 1, 0) are whole numbers and coefficients are real. [¾ mark]

p(x) = 2x² − 3x + 5

p(0):
p(0) = 2(0)² − 3(0) + 5 = 0 − 0 + 5 = 5 [1 mark]

p(1):
p(1) = 2(1)² − 3(1) + 5 = 2 − 3 + 5 = 4 [1 mark]

p(−1):
p(−1) = 2(−1)² − 3(−1) + 5 = 2(1) + 3 + 5 = 2 + 3 + 5 = 10 [1 mark]

(a) Definition of Polynomial:
An expression of the form a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀, where a₀, a₁, …, a_n are real numbers and n is a non-negative integer, is called a polynomial in variable x. The exponents of the variable must be whole numbers (0, 1, 2, 3, …). [1 mark]

(b) Degree of a Polynomial:
The degree of a polynomial is the highest power (exponent) of the variable with a non-zero coefficient. For example, in 4x³ − 7x + 9, the degree is 3. [1 mark]

(c) Examples:
• Constant polynomial: p(x) = 7 (degree 0)
• Linear polynomial: p(x) = 3x + 2 (degree 1)
• Quadratic polynomial: p(x) = x² − 4x + 1 (degree 2)
• Cubic polynomial: p(x) = 2x³ − x + 5 (degree 3) [1 mark]

(d) Maximum number of zeros:
• Constant polynomial (non-zero): 0 zeros (never equals 0)
• Linear polynomial: at most 1 zero
• Quadratic polynomial: at most 2 zeros
• Cubic polynomial: at most 3 zeros
In general, a polynomial of degree n has at most n zeros. [1 mark]

(e) Can a non-zero constant polynomial have a zero?
No. A non-zero constant polynomial, say p(x) = 5, gives the value 5 for every value of x. Since 5 ≠ 0 for any x, there is no value of x for which p(x) = 0. Therefore, a non-zero constant polynomial has no zero. [1 mark]

(a) Find the zero of p(x) = −7x + 21:
−7x + 21 = 0
−7x = −21
x = −21/(−7) = 3 [1 mark]

Verification:
p(3) = −7(3) + 21 = −21 + 21 = 0 ✓ [1 mark]

(b) Find the value of k:
p(x) = kx − 12, and zero is x = 4.
p(4) = 0
k(4) − 12 = 0
4k = 12
k = 12/4 = 3 [1 mark]

(c) Find x when cost = ₹500:
C(x) = 15x + 200
500 = 15x + 200
15x = 500 − 200 = 300
x = 300/15 = 20 [1 mark]

So the shopkeeper needs to make 20 items for the daily cost to be ₹500. [1 mark]

Yes, every real number is a zero of the zero polynomial. [1 mark]

Explanation:
The zero polynomial is p(x) = 0. For any real number a, we get p(a) = 0. Since the value is always 0, every real number satisfies p(a) = 0. Hence, every real number is a zero of the zero polynomial.

Difference from a non-zero constant polynomial:
For p(x) = 5 (a non-zero constant polynomial), p(a) = 5 ≠ 0 for every real number a. So p(x) = 5 has no zeros at all. This is the exact opposite — the zero polynomial has infinitely many zeros (every real number), while a non-zero constant polynomial has no zeros. [1 mark]