Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(x^{2}+9\right)\left(x^{2}-4\right) $$

Step-by-Step Solution

Answer

The polynomial expression is \(x^4 + 5x^2 - 36\).

Step 1: Identify the Expression

Recognize that you need to multiply two binomials: \( (x^2 + 9) \) and \( (x^2 - 4) \).

Step 2: Apply the Distributive Property (FOIL Method)

Use the distributive property to expand the expression. Multiply each term in the first binomial by each term in the second binomial: \((x^2 + 9) \cdot (x^2 - 4) = x^2 \times x^2 + x^2 \times (-4) + 9 \times x^2 + 9 \times (-4)\).

Step 3: Calculate Each Product

Work out each multiplication: \(x^2 \times x^2 = x^4\), \(x^2 \times (-4) = -4x^2\), \(9 \times x^2 = 9x^2\), and \(9 \times (-4) = -36\).

Step 4: Combine the Products to Form the Polynomial

Combine the results to write the polynomial: \(x^4 - 4x^2 + 9x^2 - 36\).

Step 5: Simplify the Polynomial Expression

Combine like terms: \(-4x^2 + 9x^2 = 5x^2\). Thus, the expression simplifies to \(x^4 + 5x^2 - 36\).

Key Concepts

Distributive property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and perform multiplication in a structured manner. It states that to multiply a sum by another number, you can multiply each addend by the number, and then add the results. When dealing with polynomials, this is extremely useful as it enables us to break down complex expressions into more manageable parts.

In our exercise, we used the distributive property to expand the multiplication of two binomials, \((x^2 + 9)(x^2 - 4)\). We distribute each term in the first binomial to every term in the second binomial:

• \(x^2 imes x^2\) results in \(x^4\)
• \(x^2 imes (-4)\) results in \(-4x^2\)
• \(9 imes x^2\) results in \(9x^2\)
• \(9 imes (-4)\) results in \(-36\)

This structured approach makes it easier to handle and combine the terms systematically.

FOIL method

The FOIL method is a specific application of the distributive property that is used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which are the pairs of terms you multiply to apply the distributive property effectively.

For the expression \((x^2 + 9)(x^2 - 4)\), the FOIL method guides the multiplication:

• **First:** Multiply the first terms of each binomial: \(x^2 imes x^2 = x^4\)
• **Outer:** Multiply the outer terms: \(x^2 imes -4 = -4x^2\)
• **Inner:** Multiply the inner terms: \(9 imes x^2 = 9x^2\)
• **Last:** Multiply the last terms of each binomial: \(9 imes -4 = -36\)

This method helps ensure that all terms are properly accounted for during multiplication.

Binomials

In algebra, a binomial is a polynomial with exactly two terms. These terms are usually separated by a plus or minus sign. Binomials play a crucial role in polynomial expressions since they are often used in problems involving multiplication, factorization, and simplification.

Our problem involves multiplying two binomials: \((x^2 + 9)\) and \((x^2 - 4)\). Each binomial has two terms:

• \(x^2 + 9\): the terms are \(x^2\) and \(9\)
• \(x^2 - 4\): the terms are \(x^2\) and \(-4\)

The multiplication of these binomials requires understanding of both their individual components and the methods such as the distributive property or FOIL method for ensuring proper multiplication.

Like terms

'Like terms' refer to terms within an algebraic expression that have the same variables raised to the same powers, though their coefficients may be different. In polynomial expressions, combining like terms is a crucial step in simplifying and solving algebraic equations.

In our expression \(x^4 - 4x^2 + 9x^2 - 36\), we identify the like terms as \(-4x^2\) and \(9x^2\). These two terms both involve \(x^2\) as a common factor and can, therefore, be combined:

• \(-4x^2 + 9x^2 = 5x^2\)

Combining like terms helps to simplify the polynomial expression, making it easier to understand and solve further equations, leading us to the simplified polynomial: \(x^4 + 5x^2 - 36\).