- Analyze the given division: \((3x^2 + 22x + 7) \div (x + 7) = 3x + 1\).
- Use the fundamental relationship: \(\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}\).
- Identify the quotient as \(3x + 1\) and the dividend as \(3x^2 + 22x + 7\).
- State the final result: \((x + 7) \cdot \boxed{3x + 1} = \boxed{3x^2 + 22x + 7}\).
Explanation
- Understanding the Division Relationship
The problem provides the division of two polynomials: \((3x^2 + 22x + 7) \div (x + 7) = 3x + 1\). In algebraic division, the relationship between the dividend, divisor, quotient, and remainder is defined by the formula: \[\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder}.\] In this specific case, the dividend is \(3x^2 + 22x + 7\), the divisor is \(x + 7\), the quotient is \(3x + 1\), and since the division results in a polynomial with no fractional part, the remainder is 0.
- Formulating the Final Equation
By substituting our identified components into the division formula \(\text{Dividend} = \text{Divisor} \times \text{Quotient} + 0\), we get: \[3x^2 + 22x + 7 = (x + 7) \cdot (3x + 1).\] This confirms the fundamental relationship that the product of the divisor and the quotient must equal the dividend. We are asked to complete the sentence in the form \((x + 7) \cdot \square = \square\), where the first box represents the quotient and the second box represents the dividend.
- Final Answer Formulation
By comparing the equation to our template, we identify that the missing expressions are \((3x + 1)\) and \((3x^2 + 22x + 7)\). Therefore, the completed sentence is \((x + 7) \cdot (3x + 1) = 3x^2 + 22x + 7\), and the missing boxes are \(\boxed{3x + 1}\) and \(\boxed{3x^2 + 22x + 7}\).
Examples
In algebra, understanding the inverse relationship between division and multiplication is like learning that if you can split a total of 20 items into 4 equal groups of 5, then reversing that logic tells you that 4 groups of 5 must make a total of 20. Similarly, if a polynomial division tells us how to split a complex expression into smaller parts, we can multiply those parts back together to verify the original expression. This concept is crucial for factoring polynomials, which is a key skill for solving complex equations and graphing curves in later studies.