CBSE Class 9 Maths Ganita Manjari Part 1 Ch 02 Introduction to Linear Polynomials Short Answer Questions with Answers

2 Mark Questions

Question 1. A student says \( 4x + 9 \) is a quadratic polynomial because it has two terms. Is the statement correct? Give reason.
Answer:
No, the statement is not correct.
The type of polynomial by degree depends on the highest power of the variable, not on the number of terms.
In \( 4x + 9 \), the highest power of x is 1, so its degree is 1.
\( \therefore 4x + 9 \) is a linear polynomial. Having two terms makes it a binomial, not a quadratic.
In simple words: Quadratic means the highest power is 2. Here the highest power is 1, so it is linear.

Exam Tip: Do not mix up the two ways of naming polynomials: by number of terms (monomial, binomial) and by degree (linear, quadratic).

Question 2. Two polynomials are \( p(x) = 2x - 10 \) and \( q(x) = 5x - 25 \). Do they have the same zero? Justify.
Answer:
Zero of p(x): \( 2x - 10 = 0 \Rightarrow x = 5 \)
Zero of q(x): \( 5x - 25 = 0 \Rightarrow x = 5 \)
\( \therefore \) Yes, both polynomials have the same zero, x = 5.
This happens because \( p(x) = 2(x - 5) \) and \( q(x) = 5(x - 5) \) share the same factor \( (x - 5) \).
In simple words: Both polynomials become zero when x is 5.

Exam Tip: Always write the justification by solving each polynomial equal to zero, not just the final number.

Question 3. A polynomial has \( p(0) = -4 \) and increases by 3 for every unit increase in x. Write p(x).
Answer:
For a linear polynomial \( p(x) = ax + b \):
p(0) = b, so b = −4.
The increase for each unit increase in x is the coefficient a, so a = 3.
\( \therefore p(x) = 3x - 4 \)
In simple words: The starting value at x = 0 gives the constant, and the steady increase gives the coefficient of x.

Exam Tip: p(0) is always the constant term of a polynomial.

Question 4. A machine follows \( M(n) = 8n - 3 \) for n = 1, 2, 3, … . A student says the constant term −3 means the first output is −3. Is this correct?
Answer:
No, the student is not correct.
The constant term −3 is the value of M(n) when n = 0, since \( M(0) = -3 \).
But the machine starts from n = 1, so the first output is \( M(1) = 8(1) - 3 = 5 \).
\( \therefore \) The first output is 5, not −3.
In simple words: The constant term tells you the value at n = 0, but this machine starts counting from n = 1.

Exam Tip: Always check where the counting starts before interpreting the constant term.

Question 5. A line \( y = 2x - 6 \) crosses the x-axis at a point. Find that point without drawing the graph.
Answer:
On the x-axis, y = 0.
\( 2x - 6 = 0 \Rightarrow 2x = 6 \Rightarrow x = 3 \)
\( \therefore \) The line crosses the x-axis at (3, 0).
In simple words: Put y equal to zero and solve for x.

Exam Tip: Write the answer as a point (3, 0), not just x = 3.

3 Mark Questions

Question 6. A delivery app uses the rule \( P(x) = 15x + 40 \) to calculate a packaging score for x items.
(a) Find P(6).
(b) State the coefficient of x and the constant term.
(c) Explain what the coefficient of x and the constant term in the rule might represent in the delivery app's packaging context.
Answer:
(a) \( P(6) = 15(6) + 40 = 90 + 40 = 130 \)
(b) Coefficient of x = 15; constant term = 40.
(c) 15 is the score added for each item packed. 40 is a fixed base score given to every order, even before any item is counted.
In simple words: Each item adds 15 points, and every order starts with 40 points.

Exam Tip: In word problems, the coefficient is the 'per item' rate and the constant is the 'fixed' part.

Question 7. A student says \( 7x - 5 \) is not a polynomial because one term is negative.
(a) Justify whether his claim is correct.
(b) State its degree.
(c) Name its constant term.
Answer:
(a) The claim is incorrect. A polynomial may have negative coefficients or constants. What it cannot have is negative or fractional powers of x. Here the powers are 1 and 0, both whole numbers, so \( 7x - 5 \) is a polynomial.
(b) The highest power of x is 1, so the degree is 1.
(c) The constant term is −5 (take the sign along with the number).
In simple words: Negative signs are allowed in polynomials. Only the powers of x must be whole numbers.

Exam Tip: When naming the constant term, always include its sign.

Question 8. The values of a quantity follow the pattern 11, 16, 21, 26, … for x = 0, 1, 2, 3, …
(a) Write a linear polynomial P(x).
(b) Find P(8).
(c) Explain how you identified the coefficient of x.
Answer:
(a) At x = 0 the value is 11, so the constant is 11. Each step increases by 5. \( P(x) = 5x + 11 \)
(b) \( P(8) = 5(8) + 11 = 40 + 11 = 51 \)
(c) The difference between consecutive values is always 16 − 11 = 21 − 16 = 26 − 21 = 5. A constant increase of 5 for each unit of x means the coefficient of x is 5.
In simple words: The first value gives the constant, and the fixed jump between values gives the coefficient of x.

Exam Tip: Check the difference between more than one pair of values to confirm the pattern is linear.

Question 9. For \( P(x) = 4x - 20 \):
(a) Find its zero,
(b) Verify it by substitution, and
(c) Explain what the zero means graphically.
Answer:
(a) \( 4x - 20 = 0 \Rightarrow 4x = 20 \Rightarrow x = 5 \). The zero is 5.
(b) \( P(5) = 4(5) - 20 = 20 - 20 = 0 \) ✓
(c) The graph of \( y = 4x - 20 \) is a straight line that crosses the x-axis at the point (5, 0). The zero is the x-coordinate of that crossing point.
In simple words: The zero is the x-value where the line meets the x-axis.

Exam Tip: For part (b), show the full substitution and end with = 0.

Question 10. A polynomial \( P(x) = ax + 8 \) has zero \( x = -4 \).
(a) Find a.
(b) Write P(x).
(c) Check the zero.
Answer:
(a) Since −4 is a zero, \( P(-4) = 0 \): \( a(-4) + 8 = 0 \Rightarrow -4a = -8 \Rightarrow a = 2 \)
(b) \( P(x) = 2x + 8 \)
(c) \( P(-4) = 2(-4) + 8 = -8 + 8 = 0 \) ✓
In simple words: Put the zero into the polynomial, set it equal to zero, and solve for a.

Exam Tip: A zero of p(x) always means p(zero) = 0; use this as your first equation.

Question 11. Observe the following table:
x: 0, 1, 2, 3
P(x): 4, 7, 10, 13
(a) Find a rule of the form \( ax + b \).
(b) Find P(0).
(c) Compute P(10).
Answer:
(a) The values increase by 3 each time (7 − 4 = 10 − 7 = 13 − 10 = 3), so a = 3. When x = 0, P = 4, so b = 4. \( P(x) = 3x + 4 \)
Check: P(2) = 6 + 4 = 10 ✓, P(3) = 9 + 4 = 13 ✓
(b) \( P(0) = 3(0) + 4 = 4 \)
(c) \( P(10) = 3(10) + 4 = 34 \)
In simple words: The table grows by 3 each step and starts at 4, so the rule is 3x + 4.

Exam Tip: Always check your rule against at least two values from the table.