Problem 33 - Algebra for College Students

Question

Find a polynomial equation with real coefficients that has the given roots. $$-1,2,3$$

Step-by-Step Solution

Answer

A polynomial with the given roots is \[ x^3 - 4x^2 + x + 6 \].

Step 1: Write the roots as factors

Given the roots \(-1\), \(2\), and \(3\), express them as factors of a polynomial. The factors will be \(x - (-1)\), \(x - 2\), and \(x - 3\). So, the factors are \(x + 1\), \(x - 2\), and \(x - 3\).

Step 2: Form the polynomial

Multiply the factors together to form the polynomial equation. Start by multiplying the first two factors: \[ (x + 1)(x - 2) = x^2 - x - 2 \]

Step 3: Multiply the result by the third factor

Now multiply the resulting polynomial from Step 2 by the third factor \(x - 3\): \[(x^2 - x - 2)(x - 3)\].

Step 4: Expand the expression

Expand the expression by distributing each term: \[ x^2(x - 3) - x(x - 3) - 2(x - 3) = x^3 - 3x^2 - x^2 + 3x - 2x + 6 \].

Step 5: Simplify the polynomial

Combine like terms to simplify the polynomial: \[ x^3 - 4x^2 + x + 6 \].

Key Concepts

Real Coefficients

Polynomials with real coefficients have numbers without any imaginary parts. Real coefficients mean every number in the polynomial's terms is a real number, like 1, -2, or 3. These coefficients are important because they ensure the final polynomial is applicable to most real-world scenarios.
When you see an equation like \(x^3 - 4x^2 + x + 6\), all terms' coefficients (1, -4, 1, and 6) are real numbers.
Understanding real coefficients helps us connect mathematical solutions to tangible quantities we often encounter in life, making the solutions practical.

Roots of a Polynomial

Roots are the values of \(x\) that make the polynomial equal to zero. Given roots show where the polynomial graph intersects the x-axis.
For instance, if you know the roots \(-1\), \(2\), and \(3\), you know that plugging these values into the polynomial makes it zero:

• \(f(-1) = 0\)
• \(f(2) = 0\)
• \(f(3) = 0\)

Identifying roots helps in constructing the polynomial itself, as each root relates directly to a factor (like \(x + 1\) for root \(-1\)). This connection is crucial in solving and understanding equations.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components that, when multiplied together, give the original polynomial.
For the roots \(-1\), \(2\), and \(3\), the factors are \(x + 1\), \(x - 2\), and \(x - 3\). These factors represent simpler binomial expressions.
Combining these factors gives us a more manageable form for further calculations:

• \((x + 1)(x - 2)(x - 3)\)

Each factor corresponds to one root and helps build the polynomial step by step, making complex polynomials easier to handle.

Expanding Polynomials

Expanding polynomials turns them from factored form back into a standard polynomial form by multiplying the factors together.
For our example factors \((x + 1)(x - 2)(x - 3)\), the steps are:

• Multiply \((x + 1)\) and \((x - 2)\) to get \(x^2 - x - 2\)

Then:

• Multiply \((x^2 - x - 2)\) by \((x - 3)\) to get \(x^3 - 3x^2 - x^2 + 3x - 2x + 6\)

Finally, combining like terms gives us \(x^3 - 4x^2 + x + 6\).
This process translates compounded expressions back into a single, more readable polynomial.

Simplifying Polynomials

Simplifying polynomials is about combining like terms to condense the polynomial into its simplest form.
In the expanded polynomial \(x^3 - 3x^2 - x^2 + 3x - 2x + 6\), we notice there are like terms (terms with the same power of \(x\)).
Combining these like terms step-by-step:

• Combine \(-3x^2 - x^2\) to get \(-4x^2\)
• Combine \(3x - 2x\) to get \(x\)

So, the simplified polynomial is \(x^3 - 4x^2 + x + 6\).
Simplifying not only makes the polynomial cleaner but also reveals its underlying structure clearly.