Problem 63 - Precalculus

Question

Solve the system of linear equations. Find the values of \(A, B, C,\) and \(D\) such that the following equation is true: \(x^{3}+x^{2}+2 x+3=(A x+B)\left(x^{2}+3\right)+(C x+D)\left(x^{2}+2\right)\)

Step-by-Step Solution

Answer

The values are \(A = 0\), \(B = 1\), \(C = 1\), \(D = 0\).

Step 1: Expand the Equation

Start by expanding both sides of the equation. Expand the right side: - \((A x + B)(x^2 + 3) = A x^3 + 3A x + B x^2 + 3B\)- \((C x + D)(x^2 + 2) = C x^3 + 2C x + D x^2 + 2D\)Combine these to expand: \[ (A x + B)(x^2 + 3) + (C x + D)(x^2 + 2) = A x^3 + B x^2 + 3A x + 3B + C x^3 + D x^2 + 2C x + 2D \= (A + C) x^3 + (B + D) x^2 + (3A + 2C) x + (3B + 2D) \]

Step 2: Align Terms

Write the expanded form of the polynomial on both sides of the equation and align similar terms. The left side is already given as \(x^3 + x^2 + 2x + 3\). Compare with the right side:\[ x^3 + x^2 + 2x + 3 = (A + C) x^3 + (B + D) x^2 + (3A + 2C) x + (3B + 2D) \]

Step 3: Set up Systems of Equations

For the polynomial to be equal for all values of \(x\), the coefficients of the corresponding powers of \(x\) must be equal. This gives us the following system of equations:1. \(A + C = 1\) (from the \(x^3\) terms)2. \(B + D = 1\) (from the \(x^2\) terms)3. \(3A + 2C = 2\) (from the \(x\) terms)4. \(3B + 2D = 3\) (from the constant terms)

Step 4: Solve the System of Equations

Solve these equations one at a time:1. From \(A + C = 1\), express \(C\) in terms of \(A\): \[(C = 1 - A)\]2. Substitute \(C\) in the third equation: \[(3A + 2(1 - A) = 2)\] Simplify to find \(A\): \[(3A + 2 - 2A = 2) => (A = 0)\]3. Using \(A = 0\), calculate \(C\): \[(C = 1 - 0 = 1)\]4. Solve \(B + D = 1\) to express \(D\) in terms of \(B\): \[(D = 1 - B)\]5. Substitute \(D\) in the fourth equation: \[(3B + 2(1 - B) = 3)\] Simplify to find \(B\): \[(3B + 2 - 2B = 3) \rightarrow (B = 1)\]6. Using \(B = 1\), calculate \(D\): \[(D = 1 - 1 = 0)\]

Step 5: Verify the Solution

Check that the found values satisfy all the original polynomial equations.- Verify with \(x^3\) terms: \(A + C = 0 + 1 = 1\) - Verify with \(x^2\) terms: \(B + D = 1 + 0 = 1\) - Verify with \(x\) terms: \(3A + 2C = 3 \times 0 + 2 \times 1 = 2\)- Verify with constant terms: \(3B + 2D = 3 \times 1 + 2 \times 0 = 3\)All conditions are satisfied.

Key Concepts

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. They are fundamental in mathematics and appear in various forms, such as linear, quadratic, and cubic functions. Each term in a polynomial is composed of a coefficient and a variable raised to a power. In the polynomial \(x^3 + x^2 + 2x + 3\), for example, the terms are:

• \(x^3\) with coefficient 1
• \(x^2\) with coefficient 1
• \(2x\) with coefficient 2
• The constant term 3

These terms are aligned by their powers, known as the degree of the polynomial. The degree is determined by the highest power of the variable in the polynomial. Here, the degree is 3.

Polynomials are not only used in algebra but also in calculus, numerical analysis, and many applied fields such as physics and engineering. They are helpful in constructing equations that model real-world phenomena.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. It provides tools for handling systems of linear equations and is essential in various areas like computer science, physics, and economics.

When solving systems of linear equations, we often look for the solution that satisfies all equations simultaneously. This is exactly what we did by setting up a system of equations for the given polynomial. We needed to find values for \(A, B, C,\) and \(D\) such that the polynomial on both sides equaled for every power of \(x\).

For example, if we have the system of equations like:

• \(A + C = 1\)
• \(B + D = 1\)
• \(3A + 2C = 2\)
• \(3B + 2D = 3\)

we can use methods from linear algebra to solve it. Typical approaches include substitution, elimination, or matrix operations.

Understanding how to manipulate and solve these equations is crucial for moving forward in both simple and complex mathematical situations. It streamlines processes in disciplines requiring precise calculations and predictions.

Expanding Expressions

Expanding expressions involves removing parentheses by distributing the terms to simplify or transform them into a different form. This is an essential skill in algebra, allowing terms to be combined or arranged.When expanding an expression like \((A x + B)(x^2 + 3)\), we distribute \(A x\) and \(B\) across each term in \((x^2 + 3)\). This results in:

• \(A x^3\)
• \(B x^2\)
• \(3A x\)
• \(3B\)

Similarly, expanding the expression \((C x + D)(x^2 + 2)\) gives:

• \(C x^3\)
• \(D x^2\)
• \(2C x\)
• \(2D\)

After expanding, combining like terms allows us to simplify the expression even further, leading us to the final simplified or expanded form where all similar terms are combined.Expanding expressions is not only valuable for solving linear equations but also appears frequently in polynomial multiplication and factoring tasks. It lays the groundwork for mastering more advanced algebraic techniques.