Problem 33 - Beginning and Intermediate Algebra

Question

What binomial factors to a) \(\quad(x+4)(x-4) ?\) b) \(\quad(4+x)(4-x) ?\)

Step-by-Step Solution

Answer

a) The product of the binomial factors \((x+4)(x-4)\) is \(x^2 - 16\). b) The product of the binomial factors \((4+x)(4-x)\) is \(16 - x^2\).

Step 1: Multiply the binomial factors

To multiply the binomial factors \((x+4)\) and \((x-4)\), we can use the FOIL method (First, Outer, Inner, Last). This method helps us expand the product of two binomials.

Step 2: Apply the FOIL method

Using the FOIL method, we have: \[(x+4)(x-4) = (x \cdot x) + (x \cdot -4) + (4 \cdot x) + (4 \cdot -4)\]

Step 3: Simplify the expression

Now let's simplify the expression by performing the required multiplication operations: \[(x \cdot x) + (x \cdot -4) + (4 \cdot x) + (4 \cdot -4) = x^2 - 4x + 4x - 16\] Notice that the term \(-4x\) and \(+4x\) cancel out and we are left with: \[x^2 - 16\] Therefore, the product of the given binomial factors \((x+4)(x-4)\) is \(\boxed{x^2 - 16}\). b) \(\quad(4+x)(4-x)\)

Step 1: Multiply the binomial factors

To multiply the binomial factors \((4+x)\) and \((4-x)\), we can use the FOIL method (First, Outer, Inner, Last). This method helps us expand the product of two binomials.

Step 2: Apply the FOIL method

Using the FOIL method, we have: \[(4+x)(4-x) = (4 \cdot 4) + (4 \cdot -x) + (x \cdot 4) + (x \cdot -x)\]

Step 3: Simplify the expression

Now let's simplify the expression by performing the required multiplication operations: \[(4 \cdot 4) + (4 \cdot -x) + (x \cdot 4) + (x \cdot -x) = 16 - 4x + 4x - x^2\] Notice that the term \(-4x\) and \(+4x\) cancel out and we are left with: \[16 - x^2\] Therefore, the product of the given binomial factors \((4+x)(4-x)\) is \(\boxed{16 - x^2}\).

Key Concepts

FOIL method

The FOIL method is a simple technique used to multiply two binomials. Binomials are expressions with two terms, like

• \((a + b)\) or \((x - y)\).

FOIL stands for First, Outer, Inner, Last, which corresponds to the order in which you multiply the terms:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the binomial pair.
• Inner: Multiply the innermost terms of the binomial pair.
• Last: Multiply the last terms in each binomial.

By following this structure, you ensure that all possible pairs are multiplied. Consider the example \((x + 4)(x - 4)\), using FOIL:- First: \(x \cdot x = x^2\)- Outer: \(x \cdot -4 = -4x\)- Inner: \(4 \cdot x = 4x\)- Last: \(4 \cdot -4 = -16\)By adding all these products, you get the expanded form, \(x^2 - 4x + 4x - 16\). Notice that the inner terms \(-4x\) and \(+4x\) cancel each other out, leading to a simplified form.

Polynomial Simplification

Polynomial simplification involves reducing an expression to its simplest form. This process often involves combining like terms. Let's use the expression \(x^2 - 4x + 4x - 16\). There is an important rule:

• Like terms are terms that have the exact same variable part.

For instance, \(-4x\) and \(+4x\) are like terms because they both have \(x\) as the variable.These terms can be added or subtracted easily. Here they add together to become zero, simplifying the expression to \(x^2 - 16\).Always check your expressions for like terms, as simplifying makes it easier to work with polynomials in future steps. Simplified solutions provide clarity and eliminate unnecessary complexity, thus allowing you to see the structure or form that may reveal important characteristics of the polynomial. Keep practice this skill to tackle more complex algebraic problems.

Difference of Squares Formula

The difference of squares formula is an efficient way to multiply specific types of binomials by recognizing a pattern in their structure. The formula is:\((a + b)(a - b) = a^2 - b^2\).This formula states that if you have two binomials that differ only by the middle sign, they can be rewritten as the square of the first term, minus the square of the second term.

• Example: \((x + 4)(x - 4)\) can be seen directly as \(x^2 - 4^2 = x^2 - 16\).
• Similarly, \((4 + x)(4 - x)\) transforms into \(4^2 - x^2 = 16 - x^2\).

Learning the difference of squares formula allows for quick multiplication of binomials that meet this pattern, bypassing more lengthy expansions such as FOIL.Recognizing these patterns in polynomials can save time and simplify calculations, particularly valuable in higher-level algebra problems or when working with complex expressions.