Question: Factor the expression $ x^4 - 13x^2 + 36 $ completely over the real numbers.
["Title: How to Factor the Expression $ x^4 - 13x^2 + 36 $ Completely Over the Real Numbers", "Meta Description:\nLearn how to factor the quartic expression $ x^4 - 13x^2 + 36 $ completely over the real numbers. Step-by-step explanation with substitution, factoring techniques, and verification.", "---", "### Understanding the Expression $ x^4 - 13x^2 + 36 $", "Factoring polynomial expressions plays a vital role in algebra, and one common strategy is to simplify complex forms by reducing their degree. Here, the expression is quartic (degree 4), but it has only even powers of $ x $, making it quadratic in form. This allows us to apply a substitution technique to make the factoring manageable.", "---", "### Step 1: Substitution to Convert to a Quadratic", "Let’s let $ u = x^2 $. Then, substituting into the expression:", "$$\nx^4 - 13x^2 + 36 = u^2 - 13u + 36\n$$", "Now we have a quadratic expression in $ u $, which we can factor more easily.", "---", "### Step 2: Factor the Quadratic $ u^2 - 13u + 36 $", "We seek two numbers that multiply to $ 36 $ and add to $ -13 $.\nThese numbers are $ -9 $ and $ -4 $, because:", "$$\n-9 \ imes -4 = 36,\quad -9 + (-4) = -13\n$$", "Thus,", "$$\nu^2 - 13u + 36 = (u - 9)(u - 4)\n$$", "---", "### Step 3: Substitute Back $ u = x^2 $", "Replacing $ u $ with $ x^2 $, we obtain:", "$$\n(x^2 - 9)(x^2 - 4)\n$$", "Now each factor is a difference of squares, which can be factored further over the real numbers.", "---", "### Step 4: Factor $ x^2 - 9 $ and $ x^2 - 4 $ as Differences of Squares", "Recall the identity:\n$$\na^2 - b^2 = (a - b)(a + b)\n$$", "Apply this:", "- $ x^2 - 9 = (x - 3)(x + 3) $\n- $ x^2 - 4 = (x - 2)(x + 2) $", "---", "### Step 5: Combine All Factors", "Putting it all together:", "$$\nx^4 - 13x^2 + 36 = (x - 3)(x + 3)(x - 2)(x + 2)\n$$", "This is the complete factorization over the real numbers.", "---", "### Why This Factorization Works Over the Reals", "Each factor is linear (degree 1), and all coefficients are real. Since the original expression has real coefficients, complex numbers are unnecessary—this factorization is fully real.", "---", "### Final Answer", "$$\n\boxed{(x - 3)(x + 3)(x - 2)(x + 2)} \quad \ ext{is the complete factorization of } x^4 - 13x^2 + 36 \ ext{ over the real numbers.}\n$$", "---", "### Tips for Factoring Similar Expressions", "- Recognize even-powered polynomials as quadratic in $ x^2 $, using substitution to simplify.\n- Use known algebraic identities (difference/sum of squares) for further factoring.\n- Always verify by expanding the factored form to ensure correctness.", "---", "Keywords:\nfactor $ x^4 - 13x^2 + 36 $, polynomial factoring, factor quartic expression, factor $ x^4 - 13x^2 + 36 $, real factorization, substitution method, difference of squares, completing the square in substitution", "Also search for:\nHow to factor $ x^4 - 13x^2 + 36, step-by-step factoring, factor quartic completely, factor real quartic expressions, simplify and factor expressions algebraically."]
