Question

Multiply. $$(-6 x-1)(-6 x+1)$$

Short answer

Answer: The product of the binomials is $$36x^2 - 1$$.

Step-by-step solution

Identify the terms

The given binomials have two terms each. The first binomial has terms $$-6x$$ and $$-1$$, and the second binomial has terms $$-6x$$ and $$1$$.

Use FOIL method to multiply the terms

FOIL stands for First, Outer, Inner, and Last. Multiply the terms in this order: 1. First terms: $$(-6x)(-6x)$$ 2. Outer terms: $$(-6x)(1)$$ 3. Inner terms: $$(-1)(-6x)$$ 4. Last terms: $$(-1)(1)$$

Multiply the terms and simplify

1. First terms: $$(-6x)(-6x) = 36x^2$$ (since \((-a)(-a) = a^2\)) 2. Outer terms: $$(-6x)(1) = -6x$$ 3. Inner terms: $$(-1)(-6x) = 6x$$ (since \((-a)(-b) = ab\)) 4. Last terms: $$(-1)(1) = -1$$ Now, simplify the expression by combining like terms (in this case, the linear terms): $$36x^2 - 6x + 6x - 1 = 36x^2 - 1$$ The final answer is $$36x^2 - 1$$.

Key concepts

Multiplying Binomials

In algebra, the process of multiplying binomials is fundamental to expanding expressions. A binomial is an algebraic expression containing two terms, such as \(a + b\) or \(x - y\). When we multiply two binomials, we apply the distributive property twice or use the FOIL method, an acronym standing for First, Outer, Inner, and Last.

This method guides us in multiplying each term in the first binomial with every term in the second binomial in a specific order, ensuring that no term is left out. For example, to multiply \(a + b\) and \(c + d\), we multiply the first terms \( a \times c \), the outer terms \( a \times d \), the inner terms \( b \times c \), and the last terms \( b \times d \), combining all the products to get \( ac + ad + bc + bd\).

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

Once you have these four products, you combine like terms to simplify the expression to its final form.

Simplifying Algebraic Expressions

After multiplying binomials, simplifying the resulting algebraic expression is the next step. Simplification involves combining like terms and simplifying any constants. Like terms are terms that have the same variable raised to the same power. These can be collected together by adding or subtracting coefficients.

We should remember that similar variable terms with different exponents are not like terms and cannot be combined, such as \(x\) and \(x^2\). In our example with the expression \(36x^2 - 6x + 6x - 1\), the terms \( -6x \text{ and } +6x\) are like terms and thus can be added to simplify the expression to \(36x^2 - 1\). It's critical to always look for these opportunities to simplify, as it helps in achieving the most reduced form of an expression.

Difference of Squares

The difference of squares is a special polynomial product that appears quite frequently in algebra. It occurs when we have two terms that are squared and separated by a subtraction sign, like \(a^2 - b^2\). This pattern is important because it factors into the product of two binomials: \(a + b\) and \(a - b\).

Pattern of Difference of Squares

• If an expression is in the form \( a^2 - b^2 \), it factors into \( (a + b)(a - b) \) based on this special pattern.

Looking at our original exercise where we have a result of \(36x^2 - 1\), this is a difference of squares because \(36x^2\) is \( (6x)^2 \) and \(1\) is \( 1^2 \). So, this can be factored back into \( (6x + 1)(6x - 1) \) if needed.

The difference of squares is a powerful tool for factoring and can greatly simplify solving equations and understanding the structure of algebraic expressions.

Algebraic Terms

Algebraic expressions are composed of algebraic terms. An algebraic term is a product of numbers, variables, or both, sometimes raised to a power. In our example, \(36x^2 \) and \( -1 \) are both algebraic terms.

• Coefficient: A number multiplied by a variable in an algebraic term, such as the 36 in \(36x^2\).
• Variable: A symbol that represents an unknown value, like \(x\) in our terms.
• Exponent: The power to which a number or variable is raised, indicated by a superscript number; for example, the 2 in \(x^2\) is an exponent.

The careful manipulation of these terms, whether it involves multiplication, division, addition, or subtraction, is a significant part of solving algebraic equations and simplifying expressions. Proper understanding of algebraic terms and their operations is a cornerstone of algebraic proficiency.