Problem 115 - Algebra and Trigonometry Real Mathematics, Real People

Question

Completely factor the expression over the real numbers. $$x^{3}+5 x^{2}-2 x-10$$

Step-by-Step Solution

Answer

The complete factorization over the real numbers is \(x^{3}+5 x^{2}-2 x-10 = (x + 5)(x - \sqrt{2})(x + \sqrt{2})\).

Step 1: Apply the Rational Root Theorem

With the Rational Root Theorem, it is found that one root of the polynomial is -5 by checking possible rational roots. So \(x + 5\) is a factor of the polynomial.

Step 2: Polynomial Division

Divide \(x^{3}+5 x^{2}-2 x-10\) by \(x + 5\) using polynomial long division. The quotient is \(x^{2} - 2\).

Step 3: Factor the Quotient

The quotient \(x^{2} - 2\) is factored further into \(x - \sqrt{2}\) and \(x + \sqrt{2}\), based on the difference of two squares.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool when factoring polynomials. It suggests that any rational solution, or root, of a polynomial with integer coefficients will be a ratio of a factor of the constant term and a factor of the leading coefficient. In simple terms, it helps us find possible rational roots that might simplify the factoring process.
To use the theorem, list all the factors of the constant term and the leading coefficient of the polynomial. For example, in the polynomial \(x^3 + 5x^2 - 2x - 10\), the constant term is \(-10\) and the leading coefficient is \(1\).

• The factors of \(-10\) are \(\pm 1, \pm 2, \pm 5, \pm 10\).
• The factor of \(1\) is \(\pm 1\).

This means the potential rational roots are combinations of these factors: \(\pm 1, \pm 2, \pm 5, \pm 10\).
Testing these, we find that \(-5\) is a root. Then, \(x + 5\) is a factor of the polynomial. This is a first step in simplifying the polynomial using factorization.

Polynomial Long Division

Once a factor has been identified using the Rational Root Theorem, Polynomial Long Division can be used to divide the polynomial by this factor. The division process is very similar to long division with numbers. For our polynomial \(x^3 + 5x^2 - 2x - 10\), we perform long division by \(x + 5\).
Here's how it works:

• Divide the first term of the polynomial \(x^3\) by the first term of the divisor \(x\), which gives \(x^2\).
• Multiply \(x^2\) by \(x + 5\), resulting in \(x^3 + 5x^2\).
• Subtract \(x^3 + 5x^2\) from \(x^3 + 5x^2 - 2x - 10\), which leaves \(-2x - 10\).
• Now divide \(-2x\) by \(x\) to get \(-2\), and repeat the multiplication and subtraction process.

This will leave you with the quotient \(x^2 - 2\) and no remainder, confirming that \(x + 5\) is indeed a factor of the polynomial.

Difference of Two Squares

Once we have the quotient \(x^2 - 2\) from the polynomial long division, it is time to factor this expression further. This is related to a special factoring pattern known as the Difference of Two Squares, which is applied to expressions in the form \(a^2 - b^2 = (a - b)(a + b)\).
For the expression \(x^2 - 2\), we can see it in the form of \(a^2 - b^2\), where \(a = x\) and \(b = \sqrt{2}\). Using the difference of squares rule, it factors into \((x - \sqrt{2})(x + \sqrt{2})\).
This expression now shows the polynomial completely factored over the real numbers as \((x + 5)(x - \sqrt{2})(x + \sqrt{2})\). Using this simple pattern allows us to easily complete the factorization process and simplify our original polynomial expression efficiently.