Problem 36 - College Algebra

Question

Find each product. $$(4-3 x)(4+3 x)$$

Step-by-Step Solution

Answer

The product is \(16 - 9x^2\).

Step 1: Apply the FOIL Method

Start by applying the FOIL method. Multiply the First terms, Outer terms, Inner terms, and Last terms together. \n\nThis gives: \((4-3x)(4+3x) = 4*4 + 4*3x - 3x*4 - 3x*3x\)

Step 2: Simplify the equation

Simplify each term to obtain: \(= 16 + 12x - 12x - 9x^2\)

Step 3: Combine like terms

Combine like terms, in this case +12x and -12x, which will cancel each other. The final result is then: \(= 16 - 9x^2\)

Key Concepts

Polynomial Multiplication

When multiplying polynomials, you are essentially applying the distributive property repeatedly to find the product of two sets of terms. One commonly used technique for multiplying binomials (polynomials with two terms) is the FOIL Method, which stands for First, Outer, Inner, and Last.To multiply two binomials, you take the following steps:

• First: Multiply the first terms from each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms from each binomial.

After applying these steps, you'll have four products that you then add together. In our example, \[(4 - 3x)(4 + 3x) = 4 \times 4 + 4 \times 3x - 3x \times 4 - 3x \times 3x\]which simplifies into a single polynomial expression.

Algebraic Expressions

Algebraic expressions involve numbers, variables (such as \(x\)), and operations (like addition, subtraction, multiplication, and division). They can be simple, like \(3x\), or complex, involving multiple terms and operations. When you multiply algebraic expressions, you need to carefully expand and combine the terms. This means following the order of operations and being meticulous about distributing each part of the expression through multiplication. In the example \[(4 - 3x)(4 + 3x),\] the expression is an algebraic combination of constants and the variable \(x\). This requires careful application of multiplication, and then combining like terms. Notice how the middle terms, \[12x - 12x,\] cancel out, which often happens when using conjugate pairs like these.

Factoring Polynomials

Factoring polynomials is essentially the reverse of multiplication. It involves breaking down a polynomial into simpler terms, or factors, that when multiplied together return the original polynomial expression. In simpler terms, you are looking for two or more expressions that, when multiplied, give you the previous expression.In our exercise, \[16 - 9x^2,\] we see the product of \[(4-3x)(4+3x).\]This result is already factored as it is presented as a difference of squares. The expression \[a^2 - b^2\] can be expressed as \[(a - b)(a + b),\] hence reversing the original multiplication process. The polynomial \[16 - 9x^2\] can be factored back into \[(4 - 3x)(4 + 3x),\] demonstrating the relationship between multiplication and factoring in polynomials.