Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (x+2)^{2}(x-2)^{2} $$

Step-by-Step Solution

Answer

The polynomial is \(x^4 - 8x^2 + 16\).

Step 1: Expand each binomial square

We need to expand each binomial. Using the formula \[(a + b)^2 = a^2 + 2ab + b^2\]we find:\[(x+2)^2 = x^2 + 4x + 4\]\[(x-2)^2 = x^2 - 4x + 4\]

Step 2: Use the distributive property

Now, we multiply the results from Step 1:\[(x^2 + 4x + 4)(x^2 - 4x + 4)\]

Step 3: Multiply the first terms

Multiply \(x^2\) from the first trinomial with each term of the second trinomial:\[x^2 \cdot x^2 = x^4\]\[x^2 \cdot (-4x) = -4x^3\]\[x^2 \cdot 4 = 4x^2\]

Step 4: Multiply the middle terms

Next, multiply \(4x\) by each term in the second trinomial:\[4x \cdot x^2 = 4x^3\]\[4x \cdot (-4x) = -16x^2\]\[4x \cdot 4 = 16x\]

Step 5: Multiply the last terms

Finally, multiply \(4\) by each term in the second trinomial:\[4 \cdot x^2 = 4x^2\]\[4 \cdot (-4x) = -16x\]\[4 \cdot 4 = 16\]

Step 6: Combine like terms

Combine all the terms obtained from the previous steps:- Terms for \(x^4\): \(x^4\)- Terms for \(x^3\): \(-4x^3 + 4x^3 = 0\)- Terms for \(x^2\): \(4x^2 - 16x^2 + 4x^2 = -8x^2\)- Terms for \(x\): \(16x - 16x = 0\)- Constant term: \(16\)So, the polynomial is:\[x^4 - 8x^2 + 16\]

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that helps to expand expressions that are raised to a power, particularly those where the expression is a binomial, or a mathematical expression with two terms. In simple terms, when you see something like

• \[(a+b)^2\]

you can expand it using the formula:

• \[(a+b)^2 = a^2 + 2ab + b^2\]

This formula shows you how a binomial will look if you expand it into a polynomial.
Applying this to our problem, we used the theorem to expand

• \((x+2)^2\)
• and \((x-2)^2\).

For

• \((x+2)^2\),

the result is

• \(x^2 + 4x + 4\),

while

• \((x-2)^2\)

expands to

• \(x^2 - 4x + 4\).

Once these expansions are done, the next steps in our solution can proceed using these expanded forms. This illustrates how the binomial theorem efficiently transforms power expressions into easier polynomial forms.

Distributive Property

The distributive property is a basic property of numbers and algebraic expressions. It allows you to multiply a term outside a parenthesis by each term inside the parenthesis, effectively distributing the multiplication across terms. In our exercise, this concept is crucial for multiplying the two expanded binomials and further simplifying them.With the polynomials

• \((x^2 + 4x + 4)\)
• and \((x^2 - 4x + 4)\),

applying the distributive property means multiplying each term in the first trinomial with every term in the second trinomial. Let's break it down step-by-step:

• First, multiply the first term \(x^2\) in the first polynomial with every term in the second. This gives:
  • \(x^2 \cdot x^2 = x^4\)
  • \(x^2 \cdot (-4x) = -4x^3\)
  • \(x^2 \cdot 4 = 4x^2\)
• Repeat this for the other terms \(4x\)
• and \(4\) in the first polynomial:
  • \(4x\) terms multiply to yield
    • \(4x^3\)
    • \(-16x^2\)
    • \(16x\).
  • \(4\) terms multiply to yield
    • \(4x^2\)
    • \(-16x\)
    • \(16\).

The distributive property simplifies the complex product of these expressions into manageable pieces, preparing them for the final step of combining like terms.

Combining Like Terms

Once all terms have been multiplied, the final key to simplifying the expression into a polynomial is combining like terms. Like terms are parts of the polynomial that have the same variable raised to the same power. Mixing them properly is essential to simplify expressions fully.For example, in our problem, once we multiplied terms from each binomial, we ended up with:

• Terms involving \(x^4: x^4\)
• Terms involving \(x^3: -4x^3 + 4x^3\)
• Terms with \(x^2: 4x^2 - 16x^2 + 4x^2\)
• Terms with \(x: 16x - 16x\)
• Constant term: \(16\)

Combining like terms means adding or subtracting their coefficients. In our case:

• The \(x^3\) terms cancel out, resulting in \(0\).
• The \(x^2\) terms combine to \(-8x^2\).
• The \(x\) terms also cancel out to give \(0\).
• The final polynomial becomes: \(x^4 - 8x^2 + 16\).

Combining like terms is the concluding step that transforms a long expression into a clear, concise polynomial, making our results neat and easy to interpret.