Problem 135 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The cube root of \(-8\) is not a real number.

Step-by-Step Solution

Answer

The original statement \(x^{2}+36=(x+6)^{2}\) is false. The statement that would make it true is \(x^{2}+36 = x^{2} + 36\).

Step 1: Evaluate statement

To evaluate the truth of the statement, first, expand the right hand side and compare with the left hand side: \((x+6)^{2} = x^{2}+12x+36\)

Step 2: Comparison with LHS

Compare LHS \(x^{2}+36\) with \(x^{2}+12x+36\) from RHS, it can be observed that the RHS has an extra term \(12x\) which isn't present in LHS.

Step 3: Make Changes to yield True Statement

The necessary change to produce a true statement is to remove the term \(12x\) from the RHS. So we need to isolate \(x^{2}\) and \(36\) on the right side. Hence, we have the true statement: \(x^{2}+36 = x^{2} + 36\)

Key Concepts

Expanding Binomials

In mathematics, expanding binomials is a common technique used to simplify expressions that involve binomials, which are algebraic expressions containing two terms. When you expand a binomial, you convert it into a polynomial by multiplying it out fully.
To perform binomial expansion, follow these steps:

• Identify the binomial you need to expand. In the example above, this binomial is \((x+6)^2\).
• Use the formula \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = x\) and \(b = 6\).
• Substitute \(a\) and \(b\) into the formula: \((x+6)^2 = x^2 + 2 \cdot x \cdot 6 + 6^2 = x^2 + 12x + 36\).

After expanding the binomial, you obtain \(x^2 + 12x + 36\). Notice how this expansion turns a simple squared binomial into a trinomial (a polynomial with three terms). Understanding how to expand binomials is crucial because it allows you to simplify and solve algebraic expressions easily.

Polynomial Comparison

Once you have expanded a binomial into a polynomial, it is important to compare it with another polynomial to check if they are equivalent. This is often needed to verify or refute algebraic statements or equations.
Consider comparing polynomials from both sides of an equation. In the given problem, the left-hand side (LHS) is \(x^2+36\), and the right-hand side (RHS) after expansion is \(x^2+12x+36\). To compare these:

• Look at each term in the polynomial.
• Compare corresponding terms on both sides. For this equation, the LHS has \(x^2\) and \(36\), while the RHS has the same terms plus an extra \(12x\).

If there are extra or missing terms, the polynomials are not equal. In our example, because \(x^2+12x+36\) has an additional term \(12x\), compared to \(x^2+36\), the initial equation is false. This comparison technique highlights disparities and helps identify necessary modifications to make the equations true.

True and False Statements in Algebra

In algebra, determining whether a statement is true or false is a vital process that assures the correctness of expressions and equations. Consider the statement: \(x^2+36=(x+6)^2\). This statement is assessed to see if both sides are equal.You start by following these steps:

• Expand the binomial on the RHS to \(x^2+12x+36\).
• Compare it with the LHS, which is \(x^2+36\).
• Since \(12x\) is present in the RHS but not in the LHS, the original statement is false.

To correct this false statement and make it true, adjust the equation so both sides match exactly. Here, removing \(12x\) from the RHS results in \(x^2+36=x^2+36\), a now true statement.Understanding how to evaluate true and false statements allows you to verify equations and make accurate algebraic judgments. This skill is key in solving complex equations and understanding mathematical reasoning.