Read the following passage and answer the given questions. Two algebraic expressions are given as (I) $6x^2 + 29x + P$ and (II) $3x + 1$. (I) is divided by (II) leaving $(2x + 9)$ as quotient and 19 as remainder. Find the value of P.

### Concept & Polynomial Division Theorem The division algorithm for polynomials states that when a dividend is divided by a divisor, it yields a quotient and a remainder. $$ \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} $$ ### Step-by-Step Solution * Given Dividend = $6x^2 + 29x + P$ * Given Divisor = $3x + 1$ * Given Quotient = $2x + 9$ * Given Remainder = $19$ * Applying the theorem: $6x^2 + 29x + P = (3x + 1)(2x + 9) + 19$ * Expand the right side: $(3x \times 2x) + (3x \times 9) + (1 \times 2x) + (1 \times 9) + 19$ * Simplify: $6x^2 + 27x + 2x + 9 + 19$ * Combine terms: $6x^2 + 29x + 28$ * Comparing both sides: $6x^2 + 29x + P = 6x^2 + 29x + 28$ * Therefore, $P = 28$. ### Exam Strategy & Shortcut To find the constant term $P$ quickly, you only need to multiply the constant terms of the divisor and quotient, and then add the remainder. The $x$ terms are irrelevant for finding $P$. Constant of divisor (1) multiplied by constant of quotient (9) plus remainder (19) gives $1 \times 9 + 19 = 28$. ### Common Pitfall A frequent mistake is forgetting to add the remainder to the product of the divisor and quotient before comparing the coefficients. ### Final Answer Therefore, the correct answer is **28**.