Find the Product (x-2)(3x+4)

TE: Find the Product (x-2)(3x+4)

The given problem is asking you to multiply two binomials using algebraic methods to find their product. The binomials in question are (x-2) and (3x+4), and you are expected to apply the distributive property (also known as the FOIL method for binomials) to find the resulting polynomial.

$(x - 2) (3 x + 4)$

Answer

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Solution:

Step 1: Apply the FOIL Method to Expand the Expression

Multiply the terms in the binomials $(x - 2) (3 x + 4)$ .

Step 1.1: Distribute the First Term

Distribute $x$ across $(3 x + 4)$ to get $x (3 x) + x (4)$ .

Step 1.2: Distribute the Second Term

Distribute $- 2$ across $(3 x + 4)$ to get $- 2 (3 x) - 2 (4)$ .

Step 1.3: Combine the Distributed Terms

Combine the results from distributing $x$ and $- 2$ to get $x (3 x) + x (4) - 2 (3 x) - 2 (4)$ .

Step 2: Simplify the Expression

Combine like terms and simplify the expression.

Step 2.1: Simplify Each Term Individually

Simplify each term resulting from the distribution.

Step 2.1.1: Apply the Commutative Property

Rearrange the terms using the commutative property to get $3 x (x) + 4 (x) - 6 (x) - 8$ .

Step 2.1.2: Multiply the Variables

Multiply $x$ by $x$ to get $x^{2}$ .

Step 2.1.2.1: Position the Variable

Position $x$ next to itself to get $3 (x \cdot x) + 4 x - 6 x - 8$ .

Step 2.1.2.2: Perform the Multiplication

Multiply $x$ by $x$ to get $3 x^{2} + 4 x - 6 x - 8$ .

Step 2.1.3: Rearrange the Coefficient

Move the coefficient $4$ in front of $x$ to get $3 x^{2} + 4 x - 6 x - 8$ .

Step 2.1.4: Multiply the Coefficients

Multiply $3$ by $- 2$ to get $3 x^{2} + 4 x - 6 x - 8$ .

Step 2.1.5: Perform the Final Multiplication

Multiply $- 2$ by $4$ to get $3 x^{2} + 4 x - 6 x - 8$ .

Step 2.2: Combine Like Terms

Combine $4 x$ and $- 6 x$ to get the final simplified expression $3 x^{2} - 2 x - 8$ .

Knowledge Notes:

The problem involves multiplying two binomials, which is a basic algebraic operation. The process used to solve this problem is known as the FOIL method, which stands for First, Outer, Inner, Last. This refers to the order in which you multiply the terms in the binomials:

First: Multiply the first terms in each binomial.
Outer: Multiply the outer terms in the product.
Inner: Multiply the inner terms.
Last: Multiply the last terms in each binomial.

After applying the FOIL method, the distributive property is used to multiply each term in the first binomial by each term in the second binomial. The distributive property states that $a (b + c) = a b + a c$ .

Once the terms are distributed, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variables raised to the same power. In this case, terms with $x^{2}$ are combined together, and terms with $x$ are combined together.

The commutative property of multiplication is used to rearrange terms. This property states that the order of multiplication does not affect the product, i.e., $a b = b a$ .

Finally, the expression is simplified by performing the multiplication and combining like terms to reach the final product. In this case, the final product of the expression $(x - 2) (3 x + 4)$ is $3 x^{2} - 2 x - 8$ .