GSEB Class 8 Maths Solutions Chapter 14 Factorization Exercise 14.4

Question 1. Find and correct the errors in the following mathematical statements?

Question 1.1. \( 4(x - 5) = 4x - 5 \)
Answer: The provided statement is incorrect. The accurate statement is \( 4(x - 5) = 4x - 20 \) because \( 4 \times 5 = 20 \).
In simple words: The original math problem has a mistake. The right answer for \( 4 \) times \( (x \) minus \( 5) \) is \( 4x \) minus \( 20 \).

Exam Tip: Remember to distribute the outside number to every term inside the parentheses when multiplying.

Question 1.2. \( x(3x + 2) = 3x^2 + 2 \)
Answer: This is an incorrect statement. The correct statement is \( x(3x + 2) = 3x^2 + 2x \). The variable \( x \) must be multiplied with every term inside the bracket.
In simple words: The first statement is wrong. The correct one is \( x(3x + 2) = 3x^2 + 2x \), where \( x \) multiplies both parts inside the bracket.

Exam Tip: Always apply the distributive property by multiplying the monomial outside the bracket by each term inside the bracket.

Question 1.3. \( 2x + 3y = 5xy \)
Answer: This is an incorrect statement. Since \( 2x \) and \( 3y \) are not like terms, they cannot be added together to form a single term. The correct statement is \( 2x + 3y = 2x + 3y \).
In simple words: You cannot add \( 2x \) and \( 3y \) because they are different kinds of terms. The correct way to write it is just \( 2x + 3y \).

Exam Tip: Only like terms (terms with the same variables raised to the same powers) can be added or subtracted.

Question 1.4. \( x + 2x + 3x = 5x \)
Answer: This is an incorrect statement. When you add \( x \), \( 2x \), and \( 3x \), you get \( 6x \). So, the correct statement is \( x + 2x + 3x = 6x \).
In simple words: Adding \( x \), \( 2x \), and \( 3x \) gives you \( 6x \). The problem's answer \( 5x \) is wrong.

Exam Tip: Combine like terms by adding their coefficients while keeping the variable part the same.

Question 1.5. \( 5y + 2y + y - 7y = 0 \)
Answer: This is an incorrect statement. If you combine the terms, \( 5y + 2y + y \) equals \( 8y \). Then, \( 8y - 7y \) equals \( y \). Therefore, the correct statement is \( 5y + 2y + y - 7y = y \).
In simple words: The original statement is incorrect. If you combine \( 5y \), \( 2y \), and \( y \), you get \( 8y \). Then, taking away \( 7y \) leaves you with just \( y \).

Exam Tip: Group and combine all positive like terms first, then subtract the negative like terms to simplify the expression accurately.

Question 1.6. \( 3x + 2x = 5x^2 \)
Answer: This is an incorrect statement. When you add \( 3x \) and \( 2x \), the result is \( 5x \), not \( 5x^2 \). Adding like terms only changes the coefficient, not the exponent. The correct statement is \( 3x + 2x = 5x \).
In simple words: The statement is wrong. Adding \( 3x \) and \( 2x \) simply makes \( 5x \), you don't change the power of \( x \).

Exam Tip: Remember that adding or subtracting like terms involves only their coefficients; the variable and its exponent remain unchanged.

Question 1.7. \( (2x)^2 + 4(2x) + 7 = 2x^2 + 8x + 7 \)
Answer: The provided statement is incorrect. For \( (2x)^2 \), both the coefficient and the variable are squared, so \( (2x)^2 = 4x^2 \). Thus, the correct statement is \( (2x)^2 + 4(2x) + 7 = 4x^2 + 8x + 7 \).
In simple words: The first statement is wrong because \( (2x)^2 \) should be \( 4x^2 \), not \( 2x^2 \). The number \( 2 \) also gets squared.

Exam Tip: When squaring a term like \( (ax)^2 \), remember to square both the coefficient \( a \) and the variable \( x \), resulting in \( a^2x^2 \).

Question 1.8. \( (2x)^2 + 5x = 4x + 5x = 9x^2 \)
Answer: This is an incorrect statement. First, \( (2x)^2 \) becomes \( 4x^2 \). Then, you cannot add \( 4x^2 \) and \( 5x \) because they are not like terms. The correct statement is \( (2x)^2 + 5x = 4x^2 + 5x \).
In simple words: The statement is wrong. \( (2x)^2 \) is \( 4x^2 \). You cannot add \( 4x^2 \) and \( 5x \) because they are different kinds of terms.

Exam Tip: Always simplify squared terms first and then check if the resulting terms are like terms before attempting addition or subtraction.

Question 1.9. \( (3x + 2)^2 = 3x^2 + 6x + 4 \)
Answer: The provided statement is incorrect. Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we get \( (3x + 2)^2 = (3x)^2 + 2(3x)(2) + (2)^2 \), which simplifies to \( 9x^2 + 12x + 4 \). Therefore, the correct statement is \( (3x + 2)^2 = 9x^2 + 12x + 4 \).
In simple words: The first statement is wrong. To square \( (3x + 2) \), you have to square \( 3x \) to get \( 9x^2 \), multiply \( 2 \), \( 3x \), and \( 2 \) to get \( 12x \), and square \( 2 \) to get \( 4 \).

Exam Tip: Always expand binomial squares using the formula \( (a+b)^2 = a^2 + 2ab + b^2 \) to ensure all terms are correctly calculated.

Question 1.10. Substituting \( x = -3 \) in:

Question 1.10.(a) \( x^2 + 5x + 4 \) gives \( (-3)^2 + 5(-3) + 4 = 9 + 2 + 4 = 15 \)
Answer: This is an incorrect statement. When \( x = -3 \) is substituted into the expression \( x^2 + 5x + 4 \), we get \( (-3)^2 + 5(-3) + 4 = 9 - 15 + 4 \). This simplifies to \( (9 + 4) - 15 = 13 - 15 = -2 \). Therefore, the correct value is \( -2 \).
In simple words: The calculation for \( x^2 + 5x + 4 \) with \( x = -3 \) is wrong. It should be \( 9 - 15 + 4 \), which equals \( -2 \).

Exam Tip: Be very careful with signs when substituting negative values into expressions, especially for squared terms and multiplication.

Question 1.10.(b) \( x^2 - 5x + 4 \) gives \( (-3)^2 - 5(-3) + 4 = 9 - 15 + 4 = -2 \)
Answer: This is an incorrect statement. Substituting \( x = -3 \) into \( x^2 - 5x + 4 \) yields \( (-3)^2 - 5(-3) + 4 = 9 + 15 + 4 \). This sum equals \( 28 \). Therefore, the correct value is \( 28 \).
In simple words: The calculation for \( x^2 - 5x + 4 \) with \( x = -3 \) is wrong. It should be \( 9 + 15 + 4 \), which adds up to \( 28 \).

Exam Tip: Pay close attention to the product of two negative numbers, which always results in a positive number (e.g., \( -5 \times -3 = +15 \)).

Question 1.10.(c) \( x^2 + 5x \) at \( x = -3 \) is \( (-3)^2 + 5(-3) = 9 - 15 = -6 \)
Answer: The provided statement is correct. When \( x = -3 \) is substituted into \( x^2 + 5x \), we get \( (-3)^2 + 5(-3) = 9 - 15 \). This simplifies to \( -6 \), which matches the statement.
In simple words: The calculation for \( x^2 + 5x \) with \( x = -3 \) is correct. It's \( 9 - 15 \), which truly is \( -6 \).

Exam Tip: Double-check the signs of each term after substitution, especially when adding or subtracting positive and negative numbers.

Question 1.11. \( (y - 3)^2 = y^2 - 9 \)
Answer: The provided statement is incorrect. Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \), we get \( (y - 3)^2 = y^2 - 2(y)(3) + (3)^2 \), which simplifies to \( y^2 - 6y + 9 \). Therefore, the correct statement is \( (y - 3)^2 = y^2 - 6y + 9 \).
In simple words: The first statement is wrong. To square \( (y - 3) \), you have to get \( y^2 \), then subtract \( 6y \), and finally add \( 9 \).

Exam Tip: Be careful not to forget the middle term \( -2ab \) when expanding \( (a-b)^2 \); it's a common mistake to just square the two terms.

Question 1.12. \( (z + 5)^2 = z^2 + 25 \)
Answer: The provided statement is incorrect. Using the identity \( (a + b)^2 = a^2 + 2ab + b^2 \), we find that \( (z + 5)^2 = z^2 + 2(z)(5) + (5)^2 \), which simplifies to \( z^2 + 10z + 25 \). Therefore, the correct statement is \( (z + 5)^2 = z^2 + 10z + 25 \).
In simple words: The first statement is wrong. When you square \( (z + 5) \), you get \( z^2 \), then add \( 10z \), and finally add \( 25 \).

Exam Tip: Always include the middle term \( 2ab \) when expanding a binomial squared, like \( (a+b)^2 \).

Question 1.13. \( (2a + 3b)(a - b) = 2a^2 - 3b^2 \)
Answer: The provided statement is incorrect. To multiply \( (2a + 3b)(a - b) \), we distribute each term: \( a(2a + 3b) - b(2a + 3b) = 2a^2 + 3ab - 2ab - 3b^2 \). This simplifies to \( 2a^2 + ab - 3b^2 \). Therefore, the correct statement is \( (2a + 3b)(a - b) = 2a^2 + ab - 3b^2 \).
In simple words: The first statement is wrong. When you multiply \( (2a + 3b) \) by \( (a - b) \), you get \( 2a^2 \), plus \( ab \), and then minus \( 3b^2 \).

Exam Tip: Use the FOIL method (First, Outer, Inner, Last) or distributive property carefully to multiply two binomials and combine any like terms at the end.

Question 1.14. \( (a + 4)(a + 2) = a^2 + 8 \)
Answer: The provided statement is incorrect. To multiply \( (a + 4)(a + 2) \), we apply the distributive property: \( a(a + 4) + 2(a + 4) = a^2 + 4a + 2a + 8 \). This simplifies to \( a^2 + 6a + 8 \). Therefore, the correct statement is \( (a + 4)(a + 2) = a^2 + 6a + 8 \).
In simple words: The first statement is wrong. When you multiply \( (a + 4) \) by \( (a + 2) \), you get \( a^2 \), plus \( 6a \), and then plus \( 8 \).

Exam Tip: Remember that multiplying binomials typically results in three terms (a quadratic expression), not just two, because of the middle terms that combine.

Question 1.15. \( (a - 4)(a - 2) = a^2 - 8 \)
Answer: The provided statement is incorrect. To multiply \( (a - 4)(a - 2) \), we apply the distributive property: \( a(a - 2) - 4(a - 2) = a^2 - 2a - 4a + 8 \). This simplifies to \( a^2 - 6a + 8 \). Therefore, the correct statement is \( (a - 4)(a - 2) = a^2 - 6a + 8 \).
In simple words: The first statement is wrong. When you multiply \( (a - 4) \) by \( (a - 2) \), you get \( a^2 \), then minus \( 6a \), and finally plus \( 8 \).

Exam Tip: Be careful with the signs when multiplying negative terms; a negative times a negative equals a positive.

Question 1.16. \( \frac{3x^2}{3x^2} = 0 \)
Answer: This is an incorrect statement. Any non-zero expression divided by itself equals \( 1 \). So, \( \frac{3x^2}{3x^2} = 1 \) (assuming \( x \neq 0 \)).
In simple words: The statement is wrong. When you divide something by itself, the answer is always \( 1 \), not \( 0 \).

Exam Tip: Remember the basic rule of division: any non-zero quantity divided by itself always yields 1.

Question 1.17. \( \frac{3x^2+1}{3x^2} = 1 + 1 = 2 \)
Answer: This is an incorrect statement. To simplify the expression, we can write \( \frac{3x^2+1}{3x^2} = \frac{3x^2}{3x^2} + \frac{1}{3x^2} \). This simplifies to \( 1 + \frac{1}{3x^2} \). Therefore, the correct statement is \( \frac{3x^2+1}{3x^2} = 1 + \frac{1}{3x^2} \).
In simple words: The statement is wrong. You can split the fraction into two parts. One part becomes \( 1 \), and the other part stays as \( \frac{1}{3x^2} \).

Exam Tip: When a sum is in the numerator, divide each term in the numerator by the denominator separately to simplify correctly.

Question 1.18. \( \frac{3x}{3x+2} = \frac{1}{2} \)
Answer: This is an incorrect statement. You cannot cancel terms that are part of a sum or difference in a fraction. The expression \( \frac{3x}{3x+2} \) cannot be simplified further. The correct statement is \( \frac{3x}{3x+2} = \frac{3x}{3x+2} \).
In simple words: The statement is wrong. You cannot just cancel the \( 3x \) from the top and bottom because of the \( +2 \) at the bottom. The fraction stays as it is.

Exam Tip: Never cancel individual terms from a numerator and denominator if they are connected by addition or subtraction; only factors can be canceled.

Question 1.19. \( \frac{3}{4x+3} = \frac{1}{4x} \)
Answer: This is an incorrect statement. Similar to the previous case, you cannot cancel terms that are part of a sum in a fraction. The expression \( \frac{3}{4x+3} \) cannot be simplified further. The correct statement is \( \frac{3}{4x+3} = \frac{3}{4x+3} \).
In simple words: The statement is wrong. You cannot cancel out the \( 3 \) because the \( +3 \) on the bottom stops it. The fraction remains unchanged.

Exam Tip: Be vigilant against canceling terms prematurely in fractions where addition or subtraction is present; always verify factors before simplification.

Question 1.20. \( \frac{4x+5}{4x} = 5 \)
Answer: This is an incorrect statement. To simplify, we should split the fraction: \( \frac{4x+5}{4x} = \frac{4x}{4x} + \frac{5}{4x} \). This simplifies to \( 1 + \frac{5}{4x} \). Therefore, the correct statement is \( \frac{4x+5}{4x} = 1 + \frac{5}{4x} \).
In simple words: The statement is wrong. You must split the fraction into two parts. \( \frac{4x}{4x} \) becomes \( 1 \), and \( \frac{5}{4x} \) stays as it is.

Exam Tip: When a sum or difference is in the numerator, remember to divide each term by the denominator individually before simplifying.

Question 1.21. \( \frac{7x+5}{5} = 7x \)
Answer: This is an incorrect statement. To simplify, we should split the fraction: \( \frac{7x+5}{5} = \frac{7x}{5} + \frac{5}{5} \). This simplifies to \( \frac{7x}{5} + 1 \). Therefore, the correct statement is \( \frac{7x+5}{5} = \frac{7x}{5} + 1 \).
In simple words: The statement is wrong. You need to divide both \( 7x \) and \( 5 \) by \( 5 \). This makes it \( \frac{7x}{5} \) plus \( 1 \).

Exam Tip: When dividing an expression with multiple terms in the numerator, ensure that each term is divided by the denominator.