UP Board Solutions Class 9 Maths Chapter 5 Polynomial and their Factors Ex 5.2

Ex 5.2 Polynomial And Their Factors अतिलघु उत्तरीय प्रश्न (Very Short Answer Type Questions)

Question 1. बहुपद x6 + x6 + 5x2 + 7 की घात ज्ञात कीजिए।
Answer: बहुपद x6 + 5x2 + 7 की घात 6 है क्योंकि बहुपद की सबसे बड़ी घात 6 है।
In simple words: A polynomial's degree is the highest exponent of its variable. Here, x⁶ has the highest power, which is 6.

🎯 Exam Tip: Focus on identifying the term with the highest power of the variable to determine the polynomial's degree accurately.

Question 2. बहुपद 3x + 7 की घात क्या है?
Answer: बहुपद 3x + 7 की घात 1 है क्योंकि बहुपद की सबसे बड़ी घात 1 है।
In simple words: For the polynomial 3x + 7, the variable 'x' has an exponent of 1, making its degree 1.

🎯 Exam Tip: Remember that for linear polynomials, the highest power of the variable is always 1.

Question 3. 2 घात का बहुपद बताइए?
Answer: x² + 2x + 1 की घात 2 है।
In simple words: A polynomial with a degree of 2 means the highest power of its variable is 2, like in \(x^2 + 2x + 1\).

🎯 Exam Tip: When asked for a polynomial of a certain degree, provide an example where the highest exponent matches the given degree.

Question 4. बहुपद x²(3x4 + 7x - 5) की घात ज्ञात कीजिए।
Answer: x²(3x4 + 7x - 5) = 3x6 + 7x3 - 5x2 की घात 6 है।
In simple words: First, multiply \(x^2\) by each term inside the parenthesis. Then, identify the highest exponent in the resulting polynomial to find its degree.

🎯 Exam Tip: Always expand expressions before determining the degree of a polynomial to ensure you capture the highest possible power.

Question 5. यदि f(x) की घात = 36 तथा g(x) की घात = 20 तब f(x)+g(x) की घात ज्ञात कीजिए ।
Answer: बहुपद f(x) की घात = 36 बहुपद g(x) की घात = 20 f(x) + g(x) की घात 36 होगी क्योंकि यह घात दोनों में बड़ी घात है।
In simple words: When adding two polynomials, the degree of the sum is generally the maximum of the individual degrees, provided the leading terms don't cancel out.

🎯 Exam Tip: For polynomial addition, the degree of the resulting polynomial will be the highest degree among the polynomials being added, unless their highest degree terms cancel out.

Question 6. यदि f(x) की घात = m तथा g(x) की घात = n, m < n, तब f (x) + g(x) की घात ज्ञात कीजिए।
Answer: यदि बहुपद f(x) की घात m तथा बहुपद g(x) की घात n है। यदि m < n तब f(x) + g(x) की घात n होगी क्योंकि n, m से बड़ी घात है।
In simple words: If one polynomial's degree (n) is strictly greater than another's (m), then the degree of their sum will be the higher degree, n.

🎯 Exam Tip: When adding polynomials with different degrees, the degree of the sum is determined by the polynomial with the higher degree.

Question 7. यदि f (x) की घात = m तथा g(x) की घात = n, m > n, तब f(x) + g(x) की घात ज्ञात कीजिए।
Answer: यदि बहुपद f(x) की घात m तथा बहुपद g(x) की घात n है तथा m > n, तब f(x) + g(x) की घात m होगी क्योंकि m, n से बड़ी घात है।
In simple words: If the degree of f(x) (m) is greater than the degree of g(x) (n), the degree of their sum f(x) + g(x) will be m.

🎯 Exam Tip: In polynomial addition, if one polynomial has a higher degree, that degree will be the degree of the resulting sum.

Question 8. यदि f(x) = x7 – x + 2x² + 1 व g(x) = -x7 + x - 2 तब f(x) + g(x) की घात ज्ञात कीजिए।
Answer: f(x) = x7 – x5 + 2x2 + 1, g(x) = -x7 + x - 2
.. f(x) + g(x) = -x5 + 2x2 + x - 1 इसमें सबसे बड़ी घात 5 है । .. f(x) + g(x) की घात 5 होगी।
In simple words: When adding polynomials, carefully combine like terms. If the highest degree terms cancel out, the degree of the sum will be determined by the next highest power.

🎯 Exam Tip: Be cautious when adding polynomials; if the leading terms cancel each other out, the degree of the sum will be lower than the original highest degrees.

Question 9. दिये गये प्रश्न 8 में f(x) – g(x) की घात क्या है?
Answer: प्रश्न 8 से F(x) – g(x) = 2x7 – x5 + 2x2 - x + 3 इसमें सबसे बड़ी घात 7 है। .. f(x) – g(x) की घात 7 होगी।
In simple words: When subtracting polynomials, if the highest degree terms do not cancel, the resulting polynomial's degree will remain the same as the highest degree of the original polynomials.

🎯 Exam Tip: In polynomial subtraction, the degree often remains the same as the highest degree of the original polynomials, especially if leading terms do not cancel.

Ex 5.2 Polynomial And Their Factors लघु उत्तरीय प्रश्न - I (Short Answer Type Questions - I)

Question 10. निम्न में से प्रत्येक बहुपद की घात ज्ञात कीजिए
(i) 0
(ii) 7
(iii) x
(iv) x + 7
(v) x² + 2x +7
(vi) 4x³ + [latex]\sqrt{5} x^{2}[/latex] - 2x + [latex]\frac{5}{7}[/latex]
(vii) (x + 5)(4x + 7)
(viii) 6x(x² + 7)
Answer:
(i) 0 एक शून्य बहुपद है, इसकी कोई घात नहीं होती।
(ii) 7 एक अचर बहुपद है, इसकी घात शून्य है।
(iii) x में सबसे बड़ी घात 1 है। इसलिए इसकी घात एक है।
(iv) x + 7 में सबसे बड़ी घात 1 है। इसलिए इसकी घात एक है।
(v) x2 + 2x + 7 में x की सबसे बड़ी घात 2 है। इसलिए इसकी घात दो है।
(vi) 4x³ + [latex]\sqrt{5} x^{2}[/latex] - 2x + [latex]\frac{5}{7}[/latex] में x की सबसे बड़ी घात 3 है। इसलिए इसकी घात तीन है।
(vii) (x + 5)(4x + 7) = 4x2 + 27x + 35 में x की सबसे बड़ी घात 2 है। इसलिए इसकी घात दो है।
(viii) 6x(x² +7) = 6x2 + 42x में x की सबसे बड़ी घात 3 है। इसलिए इसकी घात तीन है।
In simple words: The degree of a polynomial is the highest power of its variable. A constant has a degree of zero, while a zero polynomial has an undefined degree.

🎯 Exam Tip: Distinguish between the degrees of constant polynomials (degree 0) and the zero polynomial (undefined degree) to avoid common mistakes.

Question 11. यदि f (x) = 2x² +3x +1 व g(x) = 0 तब f(x).g(x) ज्ञात कीजिए।
Answer: f(x) = 2x2 + 3x +1, g(x) = 0
f(x) . g(x) = (2x2 + 3x + 1) . 0 = 0
In simple words: Multiplying any polynomial by the zero polynomial (which is just 0) always results in the zero polynomial, meaning the product is 0.

🎯 Exam Tip: Any polynomial multiplied by the zero polynomial will always yield zero, regardless of the complexity of the non-zero polynomial.

Question 12. x2 – 4 को x3 – 4 बनाने के लिए क्या जोडें?
Answer: माना x2 – 4 में A जोड़ा जाए जिससे योगफल x3 – 4 हो जाए।
.. x2 – 4 + A = x3 – 4
.: A = x3 – 4 – x2 + 4
A = x² - 2
In simple words: To find what to add, subtract the original polynomial from the target polynomial. Simplify the resulting expression to get the required term.

🎯 Exam Tip: When asked 'what to add/subtract', formulate an algebraic equation where the unknown is the quantity to be added or subtracted, then solve for it.

Question 13. बहुपद x4 + 3x3 + 4x2 – 3x – 6 से 3x3 + 4x2 – x + 3 प्राप्त करने के लिए क्या घटायें।
Answer: माना x4 + 3x3 + 4x2 – 3x – 6 में से A घटाया जाए, जिससे शेषफल 3x3 + 4x2 – x + 3 प्राप्त हो जाए।
x4 + 3x3 + 4x2 – 3x – 6 – A = 3x3 + 4x2 - x +3
-A = 3x3 + 4x2 - x + 3 – x4 – 3x3 – 4x2 + 3x + 6
-A = -x4 + 2x + 9
.: A = x4 – 2x -9
.. x4 – 2x – 9 घटाया जाएगा।
In simple words: To find what to subtract, set up an equation where the original polynomial minus the unknown polynomial equals the target polynomial. Then, isolate the unknown.

🎯 Exam Tip: When solving for a 'what to subtract' problem, rearrange the equation carefully to isolate the unknown term, paying attention to sign changes.

Ex 5.2 Polynomial And Their Factors लघु उत्तरीय प्रश्न - II (Short Answer Type Questions - II)

Question 14. बहुपद x4 – x2 + x + 2 से x2 + x + 4 प्राप्त करने के लिये क्या जोडें?
Answer: माना x4 – x2 + x + 2 में A जोड़ा जाए जिससे x2 + x + 4 प्राप्त हो जाए।
x4 - x2 + x + 2 + A = x2 + x +4
.: A = x2 + x + 4 – x4 + x2 - x - 2 = -x4 + 2x2 + 2
In simple words: To determine what to add to a polynomial to get another, subtract the original polynomial from the desired polynomial.

🎯 Exam Tip: In addition problems, always align terms with the same powers of the variable before performing operations to avoid errors.

Question 15. 7x3 + x2 – 3x + 4 प्राप्त करने के लिये बहुपद 8x3 – 3x2 + 5x – 9 में क्या जोडें?
Answer: माना 8x3 - 3x2 + 5x - 9 में A जोड़ा जाए जिससे 7x3 + x2 - 3x + 4 प्राप्त हो जाए ।
8x3 - 3x2 + 5x - 9 + A = 7x3 + x2 - 3x +4
.: A = 7x3 + x2 - 3x + 4 – 8x3 + 3x2 - 5x + 9
= -x3 + 4x2 - 8x + 13
In simple words: To find what polynomial needs to be added to one polynomial to transform it into another, subtract the original polynomial from the target polynomial.

🎯 Exam Tip: Always perform polynomial subtraction by changing the signs of all terms in the subtrahend and then adding them to the minuend.

Question 16. x3 – 7x2 + 5 प्राप्त करने के लिये बहुपद x4 – x2 + 2x + 3 में क्या जोडें?
Answer: माना x4 – x2 + 2x + 3 में A जोड़ा जाए जिससे x3 – 7x2 + 5 प्राप्त हो जाए।
x4 - x2 + 2x + 3 + A = x3 - 7x2 + 5
.: A = x3 – 7x2 + 5 – x4 + x2 - 2x - 3
= -x4 + x3 - 6x2 - 2x + 2
In simple words: To find the polynomial to add, subtract the given starting polynomial from the desired resulting polynomial, then combine like terms.

🎯 Exam Tip: Ensure correct sign handling, especially when subtracting an entire polynomial, by distributing the negative sign to every term inside the parenthesis.

Question 17. -x2 + 3x3 – 2x + 4 प्राप्त करने के लिये बहुपद x2 – 3x3 + 2x – 4 में से क्या घटायें?
Answer: माना x2 – 3x3 + 2x – 4 में से A घटाया जाए जिससे –x2 + 3x3 – 2x + 4 प्राप्त हो ।
x2 – 3x3 + 2x – 4 – A = -x2 + 3x3 – 2x + 4
.. x2 – 3x3 + 2x – 4 + x2 – 3x3 + 2x - 4 = A
-6x3 + 2x2 + 4x - 8 = A
In simple words: To find what polynomial to subtract, set up the equation (original polynomial) - (unknown polynomial) = (target polynomial) and solve for the unknown.

🎯 Exam Tip: When moving terms across the equals sign, always remember to change their operation (addition to subtraction, subtraction to addition).

Question 18. संख्या 2 प्राप्त करने के लिये बहुपद x3 – 3x2 + 2x – 1 से क्या घटायें?
Answer: माना x3 – 3x2 + 2x – 1 में से A घटाया जाए जिससे शेषफल 2 प्राप्त हो ।
x3 - 3x2 + 2x - 1 - A = 2
x3 - 3x2 + 2x - 1 - 2 = A
x3 - 3x2 + 2x - 3 = A
In simple words: To get a constant (like 2) from a polynomial by subtraction, arrange the equation and isolate the polynomial to be subtracted, then simplify.

🎯 Exam Tip: Treat constants as polynomials of degree zero; ensure careful arithmetic when combining them with other polynomial terms.

Question 19. संख्या 1 प्राप्त करने के लिये बहुपद x3 – 2x2 + 4x + 1 से क्या घटायें?
Answer: माना x3 – 2x2 + 4x + 1 में से A घटाया जाए जिससे शेषफल 1 प्राप्त हो ।
x3 – 2x2 + 4x + 1 - A = 1
x3 - 2x2 + 4x + 1 - 1 = A
x3 - 2x2 + 4x = A
In simple words: To find what polynomial to subtract to get a specific constant, subtract that constant from the original polynomial and simplify.

🎯 Exam Tip: When dealing with subtraction to achieve a constant, remember that constants are just terms with a variable raised to the power of zero.

Question 20. बहुपद x4 – 3x3 + 2x + 6 व x4 – 3x2 + 6x + 2 के योग मे से x3 – 3x + 4 घटाइये।
Answer: योगफल = x4 – 3x3 + 2x + 6 + x4 – 3x2 + 6x + 2 = 2x4 – 3x3 – 3x2 + 8x + 8
अन्तर = 2x4 – 3x3 – 3x2 + 8x + 8 – (x3 – 3x + 4)
= 2x4 – 3x3 – 3x2 + 8x + 8 - x3 + 3x - 4 = 2x4 – 4x3 – 3x2 + 11x + 4
In simple words: First, find the sum of the two polynomials by combining like terms. Then, subtract the third polynomial from this sum, remembering to distribute the negative sign.

🎯 Exam Tip: Always perform operations in the correct order (addition/subtraction of polynomials, then further subtraction if required) and pay close attention to signs during distribution.

Question 21. बहुपद x2 – 3x3 + 2x + 5 व 2x4 – 3x3 + 9x +12 के योग में से 5x3 – 3x2 + 8 घटाइये ।
Answer: योगफल = x2 – 3x3 + 2x + 5 + 2x4 – 3x3 + 9x + 12 = 3x4 – 6x3 + 11x + 17
अन्तर = 3x4 – 6x3 + 11x + 17 – (5x3 – 3x2 + 8)
= 3x4 – 6x3 + 11x + 17 – 5x3 + 3x2 – 8
= 3x4 – 11x3 + 3x2 + 11x + 9
In simple words: First, sum the two given polynomials. Then, subtract the third polynomial from this resultant sum, combining similar terms carefully.

🎯 Exam Tip: Always arrange the polynomials in descending order of their degrees before adding or subtracting to simplify calculations and reduce errors.

Question 22. यदि p(x) = x3 – x2 + 2 तथा g(x) = x +1 तब p(x) + g(x) तथा p(x) – g(x) की घात ज्ञात कीजिए ।
Answer:
(i) p(x) = x3 – x2 + 2 तथा g(x) = x +1
p(x) + g(x) = x3 – x2 + 2 + x + 1 = x3 – x2 + x + 3
p(x) + g(x) की घात 3 है।।
(ii) p(x) – g(x) = x3 – x2 + 2 - x - 1 = x3 – x2 - x +1
p(x) – g(x) की घात 3 है।
In simple words: Add and subtract the given polynomials, then identify the highest power of the variable in each resulting expression to determine their degrees.

🎯 Exam Tip: When adding or subtracting polynomials, if the leading terms don't cancel, the degree of the resultant polynomial will be the maximum of the original degrees.

Question 23. निम्न बहुपद युग्मों का योग ज्ञात कीजिए ।
(i) 3x² + 5x - 2 ; -3x2 – 5x + 6
(ii) 3x2 -7x + 5; 6x3; + 5x - 7
(iii) x2 + x - 7; x3 + x2 + 3x + 4
(iv) x3 – 5x2 + x + 2; x3 – 3x2 + 2x + 1
(v) x6 – 3x4; x4 + x3 + 2x2 – 6
Answer:
(i) योगफल = (3x2 + 5x - 2) + (-3x2 - 5x + 6) = 4
(ii) योगफल = (3x2 - 7x + 5) + (6x3 + 5x - 7) = 6x3 + 3x2 - 2x - 2
(iii) योगफल = (x2 + x -7) + (x3 + x2 + 3x + 4) = x3 + 2x2 + 4x - 3
(iv) योगफल = (x3 – 5x2 + x + 2) + (x3 – 3x2 + 2x + 1) = 2x3 – 8x2 + 3x + 3
(v) योगफल = (x6 – 3x4) + (x4 + x3 + 2x2 – 6) = x6 – 2x4 + x3 + 2x2 – 6
In simple words: For each pair, add the polynomials by combining terms with the same variable and exponent. Be careful with positive and negative signs.

🎯 Exam Tip: When adding multiple polynomials, group like terms together first to simplify the process and reduce errors in calculation.

Question 24. निम्न में पहले बहुपद में से दूसरा बहुपद घटाइये ।
(i) x3 + x + 1; 1 - x - x2
(ii) 6x3 +5x2 + 4x – 3; 4x3 – 2x2 + 7x - 1
(iii) x3 + x2 + x + 1; x3 – x2 + x - 1
(iv) x4 – 3x3 + 2x + 6; x4 – 3x3 - 6x + 2
(v) 3x7 – 2x2 + 3; x6 – 3x4 + x2 + x
Answer:
(i) अन्तर = (x3 + x + 1) – (1 - x - x2) = x3 + x + 1 - 1 + x + x2 = x3 + x2 + 2x
= x(x2 + x + 2)
(ii) अन्तर = 6x3 + 5x2 + 4x - 3 – (4x3 – 2x2 + 7x - 1)
= 6x3 + 5x2 + 4x - 3 – 4x3 + 2x2 - 7x + 1 = 2x3 + 7x2 - 3x - 2
(iii) अन्तर = (x3 + x2 + x + 1) – (x3 – x2 + x - 1)
= x3 + x2 + x + 1 - x3 + x2 - x + 1 = 2x2 +2
(iv) अन्तर = (x4 – 3x3 + 2x + 6) – (x4 – 3x3 – 6x + 2)
= x4 – 3x3 + 2x + 6 - x4 + 3x3 + 6x - 2 = 8x + 4
(v) अन्तर = (3x7 – 2x2 + 3) – (x6 – 3x4 + x2 + x)
= 3x7 – 2x2 + 3 - x + 3x4 – x2 - x = 3x7 – x6 + 3x4 – 3x2 - x + 3
In simple words: To subtract polynomials, change the sign of each term in the second polynomial and then add it to the first polynomial, combining like terms.

🎯 Exam Tip: A common error in subtraction is forgetting to distribute the negative sign to *every* term of the polynomial being subtracted; double-check all signs.

Question 25. निम्न बहुपदों का गुणनफल ज्ञात कीजिए।
(i) x2 – 4x + 4; x - 2
(ii) x2 + 3x + 2; x2 + 3x +1
(iii) 3x3 + x2 + x; x + 2
(iv) 3x + 2; x2 + x + 1
Answer:
(i) गुणनफल = (x2 – 4x + 4)(x - 2) = x3 – 2x2 - 4x2 + 8x + 4x - 8 = x3 – 6x2 + 12x - 8
(ii) गुणनफल = (x2 + 3x + 2)(x2 + 3x + 1)
= x4 + 3x3 + x2 + 3x3 + 9x2 + 3x + 2x2 + 6x + 2
= x4 + 6x3 + 12x2 + 9x + 2
(iii) गुणनफल = (3x3 + x2 + x)(x + 2)
= 3x4 + 6x3 + x3 + 2x2 + x2 + 2x
=3x4 + 7x3 + 3x2 + 2x
(iv) गुणनफल = (3x + 2)(x2 + x + 1)
= 3x3 + 3x2 + 3x + 2x2 + 2x + 2
= 3x3 +5x2 + 5x + 2
In simple words: Multiply each term of the first polynomial by every term of the second polynomial, then combine all the resulting like terms.

🎯 Exam Tip: Use the distributive property carefully when multiplying polynomials. It's often helpful to keep track of terms by aligning them based on their degrees.