Given the quadratic expression Mx^2 + Nx + 21, if (x + 3) is a factor, how can we find the relationship between M and N?

Since (x + 3) is a factor, substituting x = -3 into the quadratic gives zero: M(-3)^2 + N(-3) + 21 = 0, which simplifies to 9M - 3N + 21 = 0. This provides a relationship between M and N.

What equation do we get when we substitute x = -3 into Mx^2 + Nx + 21, given that (x + 3) is a factor?

Substituting x = -3 yields 9M - 3N + 21 = 0, which simplifies to 3M - N + 7 = 0.

Given that M and N are natural numbers, what is the smallest possible value of M + N based on N = 3M + 7?

Since M is natural, the smallest M can be is 1, which gives N = 3(1) + 7 = 10. Therefore, M + N = 1 + 10 = 11, which is the smallest possible value.

What is the smallest possible value of M + N when M and N are natural numbers and (x + 3) is a factor of Mx^2 + Nx + 21?

The smallest possible value is 11, achieved when M = 1 and N = 10.

Are there other pairs of natural numbers for M and N that satisfy the factor condition and result in a smaller M + N?

No, because increasing M increases N (since N = 3M + 7), which in turn increases M + N. The minimal sum occurs when M = 1, N = 10.

One Of The Factors Of Mx2 +nx + 21 Is X+3. Find The Smallestpossible Value Of M + N. (m Andn Are Natural

One Of The Factors Of Mx2 + nx + 21 Is X+3. Find The Smallestpossible Value Of M + N. (m And n Are Natural

Understanding algebraic factors and polynomial relationships is a fundamental aspect of mathematics. In particular, problems involving quadratic polynomials and their factors provide insight into how algebraic expressions are constructed and decomposed. Today, we'll explore a problem that requires identifying the smallest possible sum of two natural numbers, M and N, given specific conditions on a quadratic polynomial and one of its factors. This problem not only enhances problem-solving skills but also deepens understanding of polynomial factorization and the properties of natural numbers.

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Understanding the Problem Statement

Let's restate the problem to ensure clarity:

The quadratic polynomial is given as Mx² + nx + 21, where M and N are natural numbers (positive integers).

One of its factors (i.e., a polynomial that divides it exactly) is x + 3.

The goal is to find the smallest possible value of M + N.

This problem involves polynomial factorization, specifically recognizing factors and their implications on the coefficients of the quadratic polynomial.

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Key Concepts to Solve the Problem

To approach this problem, it’s essential to understand several algebraic concepts:

1. Polynomial Factorization

A quadratic polynomial can often be factored into two binomials.
If x + 3 is a factor, then the quadratic polynomial can be expressed as:

Mx² + nx + 21 = (x + 3)(A x + B)

where A and B are constants to be determined.

2. Polynomial Expansion

Expanding the factored form:

(x + 3)(A x + B) = A x² + B x + 3 A x + 3 B

Simplify:

A x² + (B + 3A) x + 3 B

Equate this to the original polynomial:

M x² + n x + 21

From the comparison, deduce relationships between A, B, M, n, and the constant term 21.

3. Coefficient Matching

Match coefficients for each power of x:

Coefficient of x²: M = A

Coefficient of x: n = B + 3A

Constant term: 21 = 3 B

4. Solving for B and A

From the constant term:

21 = 3 B → B = 21 / 3 = 7

Since B = 7, then:

n = B + 3A = 7 + 3A

Recall M = A, with A being a natural number (positive integer).

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Determining the Values of M and N

Based on the relationships:

M = A (a natural number)

N = n = 7 + 3A

Since M and N are natural numbers, and A is a natural number, the problem reduces to:

Find the smallest possible value of M + N = A + (7 + 3A) = 4A + 7

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Minimizing M + N

To find the smallest possible value of M + N:

Since A is a natural number (A ≥ 1),

Compute the sum for A = 1:

M + N = 4(1) + 7 = 4 + 7 = 11

For A = 2:

M + N = 8 + 7 = 15

For A = 3:

M + N = 12 + 7 = 19

And so on.

The smallest value occurs when A = 1:

M + N = 11

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Verifying the Solution

Let's verify whether the values M=1 and N=10 satisfy all conditions:

M = 1 (natural number)

N = 10 (since N = n = 7 + 31 = 10)

Polynomial: 1 x² + 10 x + 21 = x² + 10 x + 21

Factorization:

Polynomial factors as (x + 3)(A x + B):

With A=1, B=7:

(x + 3)(x + 7) = x² + 7x + 3x + 21 = x² + 10x + 21

which matches the original polynomial.

Since (x + 3) is a factor, the conditions are satisfied.

Thus, the minimal sum M + N is 11.

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Conclusion and Final Remarks

This problem elegantly demonstrates how polynomial factorization can be used to determine unknown coefficients given certain factors. By expressing the quadratic polynomial in factored form and matching coefficients, we derived relationships between the variables and identified the minimal sum of M and N that satisfy the conditions.

Key takeaways:

Recognizing factors of polynomials helps in reverse-engineering coefficients.

Expressing the polynomial as a product of binomials simplifies the problem.

Ensuring coefficients are natural numbers constrains the possible solutions.

The minimal sum of M + N is achieved when the associated parameter A is minimized, which in turn minimizes the sum.

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Additional Tips for Solving Similar Polynomial Problems

Always start by expressing the polynomial in its factored form when a factor is known.
Use coefficient comparison to set up equations relating unknowns.
Remember that natural numbers are positive integers; this constrains the possible values.
Verify solutions by expanding the factors to ensure they match the original polynomial.
Explore the impact of varying parameters systematically to find minimal or maximal values.

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Final Thoughts

Mathematics often involves exploring relationships and constraints to find optimal solutions. The problem of determining the smallest possible value of M + N, given a quadratic polynomial with a known factor, exemplifies this approach. By applying algebraic techniques such as factorization, coefficient matching, and systematic checking, students and enthusiasts can develop a deeper understanding of polynomial behavior and problem-solving strategies.

Whether you're preparing for exams, practicing algebra, or simply exploring mathematical relationships, mastering these techniques will serve you well in tackling a wide variety of algebraic problems with confidence and precision.