CBSE Class 10 Mathematics Polynomials Assignment Set 10

Fill in the Blanks

Question 1. The degree of a cubic polynomial is ________.
Answer:
A cubic polynomial has degree 3.
In simple words: Cubic means degree three.

Teacher's Note:
1. Linear is 1, quadratic is 2, cubic is 3.
2. Example: x³ − 2x + 1.

Question 2. A polynomial of degree n has at the most ________ zeroes.
Answer:
A polynomial of degree n has at the most n zeroes.
In simple words: The number of zeroes can never be more than the degree.

Teacher's Note:
1. A cubic can have 1, 2 or 3 real zeroes.
2. The graph meets the x-axis at most n times.

Question 3. ________ is not equal to zero when the divisor is not a factor of the dividend.
Answer:
The remainder is not zero when the divisor is not a factor of the dividend.
In simple words: A non-zero remainder means the division is not exact.

Teacher's Note:
1. Exact division means the remainder is 0.
2. Use the remainder theorem to find the remainder quickly.

Question 4. The degree of the remainder is always ________ than the degree of the divisor.
Answer:
The degree of the remainder is always less than the degree of the divisor (or the remainder is zero).
In simple words: The division stops when the remainder is of lower degree.

Teacher's Note:
1. This is the condition in the division algorithm.
2. Divide by a quadratic and the remainder has degree at most 1.

Question 5. The degree of the zero polynomial is ________.
Answer:
The degree of the zero polynomial is not defined.
In simple words: Zero has no highest power.

Teacher's Note:
1. A non-zero constant has degree 0.
2. Do not mix the two.

Question 6. A linear polynomial is represented by a ________.
Answer:
The graph of a linear polynomial is a straight line.
In simple words: ax + b gives a straight line.

Teacher's Note:
1. Two points are enough to draw it.
2. It cuts the x-axis at exactly one point.

Question 7. The highest power of a variable in a polynomial is called its ________.
Answer:
It is called the degree of the polynomial.
In simple words: The biggest exponent is the degree.

Teacher's Note:
1. Look at every term before deciding.
2. For several variables add the exponents in each term.

Question 8. The graph of a quadratic polynomial opens upward when 'a' is ________.
Answer:
The parabola opens upward when a is positive (a > 0).
In simple words: A positive coefficient of x² gives a cup shape.

Teacher's Note:
1. a < 0 gives an inverted cup.
2. a can never be 0 for a quadratic.

Question 9. A ________ is a polynomial of degree 0.
Answer:
A non-zero constant polynomial has degree 0.
In simple words: A plain number is a constant polynomial.

Teacher's Note:
1. Write 5 = 5x⁰.
2. The zero polynomial is different.

Question 10. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the ________ axis.
Answer:
The graph intersects the x-axis at the zeroes of p(x).
In simple words: Zeroes are where the curve meets the x-axis.

Teacher's Note:
1. On the x-axis y = 0.
2. So p(x) = 0 there.

True and False Statements

Question 11. 3, −1 and \( \frac{1}{3} \) are the zeroes of the cubic polynomial \( p(x) = 3x^3 - 5x^2 - 11x - 3 \).
Answer:
p(3) = 81 − 45 − 33 − 3 = 0 and p(−1) = −3 − 5 + 11 − 3 = 0, but \( p\left(\frac13\right) = \frac19 - \frac59 - \frac{11}{3} - 3 \neq 0 \).
Since p(−1/3) = −1/9 − 5/9 + 11/3 − 3 = 0, the zeroes are 3, −1 and \( -\frac13 \).
False.
In simple words: The third zero is −1/3, not 1/3.

Teacher's Note:
1. Test each number by substitution.
2. Sum of zeroes = 3 − 1 − 1/3 = 5/3 = −(−5)/3 confirms it.

Question 12. (z − 1) is a factor of g(z) = 2z³ − 2.
Answer:
g(1) = 2 − 2 = 0, so by the factor theorem (z − 1) is a factor. True.
In simple words: g(1) = 0.

Teacher's Note:
1. Apply the factor theorem.
2. 2z³ − 2 = 2(z − 1)(z² + z + 1).

Question 13. \( x^3 - 2\sqrt{x} \) is a polynomial.
Answer:
False. \( \sqrt{x} = x^{1/2} \), and the exponent of x is not a whole number.
In simple words: The power 1/2 is not allowed.

Teacher's Note:
1. Exponents must be non-negative integers.
2. √x is not a polynomial term.

Question 14. A cubic polynomial has at least one zero.
Answer:
True. The graph of a cubic polynomial always crosses the x-axis at least once, and at most three times.
In simple words: A cubic curve always meets the x-axis.

Teacher's Note:
1. Its ends go in opposite directions.
2. A quadratic may have no real zero, a cubic always has one.

Question 15. The degree of the sum of two polynomials each of degree 3 is always 3.
Answer:
False. For example (x³ + x) + (−x³ + 2) = x + 2, which has degree 1.
In simple words: The leading terms can cancel.

Teacher's Note:
1. Try opposite leading coefficients.
2. The degree of the sum is at most 3.

Question 16. The sum of the zeroes of a quadratic polynomial = − (coefficient of x) / (coefficient of x²).
Answer:
True. α + β = −b/a.
In simple words: This is the standard relation.

Teacher's Note:
1. Remember the minus sign.
2. Product of zeroes is c/a.

Question 17. The graph of a quadratic polynomial is a parabola.
Answer:
True.
In simple words: y = ax² + bx + c draws a parabola.

Teacher's Note:
1. It opens up when a > 0 and down when a < 0.
2. It has one turning point.

Question 18. The degree of the product of two polynomials each of degree 5 is always 5.
Answer:
False. Degrees add on multiplication: 5 + 5 = 10.
In simple words: Multiplying x⁵ by x⁵ gives x¹⁰.

Teacher's Note:
1. Degree of product = sum of degrees.
2. Degree of a sum is at most the larger degree.

Question 19. The product of the zeroes of a quadratic polynomial = − (constant term) / (coefficient of x²).
Answer:
False. The product of the zeroes is + (constant term) / (coefficient of x²) = c/a.
In simple words: There is no minus sign in the product.

Teacher's Note:
1. Only the sum has a minus sign (−b/a).
2. Check with x² − 5x + 6: product 6.

Question 20. The zeroes of the quadratic polynomial x² + 6x + 9 are 2 and 7.
Answer:
x² + 6x + 9 = (x + 3)², so the zeroes are −3 and −3. False.
In simple words: The zeroes are −3, −3.

Teacher's Note:
1. Factorise before guessing.
2. 2 and 7 would give the sum 9 and product 14.

Match the Following

Question 21. Match the zeroes with the quadratic polynomials.
(1) 3 and −3   (A) x² + x − 42
(2) 5 + √2 and 5 − √2   (B) x² − 9
(3) −9 and 1/9   (C) x² + (80/9)x − 1
(4) −7 and 6   (D) x² − 10x + 23
(a) 1-A, 2-C, 3-B, 4-D
(b) 1-C, 2-B, 3-D, 4-A
(c) 1-B, 2-D, 3-C, 4-A
(d) 1-D, 2-A, 3-C, 4-B
Answer:
Answer: (c) 1-B, 2-D, 3-C, 4-A
(1) Sum 0, product −9: x² − 9 (B).
(2) Sum (5 + √2) + (5 − √2) = 10, product 25 − 2 = 23: x² − 10x + 23 (D).
(3) Sum −9 + 1/9 = −80/9, product −1: x² + (80/9)x − 1 (C).
(4) Sum −1, product −42: x² + x − 42 (A).
In simple words: Find the sum and product of each pair and write x² − (sum)x + product.

Teacher's Note:
1. Do the four pairs one by one.
2. The polynomial for 5 ± √2 has constant term 23.

Question 22. Match each division with its remainder.
(1) (x³ − 3x² + x + 2) ÷ (x² − x + 1)   (A) 8
(2) (3x³ + 5x² + 2) ÷ (x² + 2x + 1)   (B) 41x − 18
(3) (x⁴ + 7x³ + 3x + 2) ÷ (x² + 3x − 2)   (C) −x + 3
(4) (x⁴ − 3x² + 4x + 5) ÷ (x² − x + 1)   (D) −2x + 4
(a) 1-D, 2-C, 3-B, 4-A
(b) 1-C, 2-B, 3-D, 4-A
(c) 1-A, 2-C, 3-D, 4-B
(d) 1-B, 2-D, 3-A, 4-C
Answer:
Answer: (a) 1-D, 2-C, 3-B, 4-A
(1) Quotient x − 2, remainder −2x + 4 (D).
(2) Quotient 3x − 1, remainder −x + 3 (C).
(3) Quotient x² + 4x − 10, remainder 41x − 18 (B).
(4) Quotient x² + x − 3, remainder 8 (A).
In simple words: Do each long division and read off the remainder.

Teacher's Note:
1. Check each by quotient × divisor + remainder.
2. The remainder has a lower degree than the divisor.