Problem 179 - College Algebra

Question

Exercises \(177-179\) will help you prepare for the material covered in the next section. If \(-8\) is substituted for \(x\) in the equation \(5 x^{3}+11 x^{3}+2-0\) is the resulting statement true or false?

Step-by-Step Solution

Answer

The statement is false, because the resulting value from Step 2 is not equal to zero.

Step 1: Substitute x with -8

Replace \(x\) in the expression \(5 x^{3}+11 x^{3}+2\) with \(-8\). So, the expression becomes \(5 (-8)^{3}+11 (-8)^{3}+2\).

Step 2: Simplify the equation

Now carry out the operations for the simplified expression. Start with the exponents to get \(-320 -704 + 2\). Then add the values to get a total.

Step 3: Check if the result is equal to 0

Now check if the resulting value from Step 2 is equal to zero or not. If it's zero, then the statement is true; if it's not zero, then the statement is false.

Key Concepts

Substitution Method

The substitution method is a powerful tool in algebra that allows us to simplify expressions by replacing variables with given values. In our exercise, we substitute the variable \( x \) with \(-8\). This is the step where we insert the specific value into the expression and prepare to evaluate it. To do this efficiently:

• Identify the variable you need to substitute.
• Replace every instance of the variable in the expression with the given value.
• It’s important to use parentheses when substituting, especially when dealing with negative numbers and exponents, to avoid mistakes.

In our example, replacing every \( x \) with \( -8 \) changes the expression from \( 5x^3 + 11x^3 + 2 \) into \( 5(-8)^3 + 11(-8)^3 + 2 \). Proper substitution sets the foundation for further calculations.

Polynomials

Polynomials are a cornerstone of algebra, consisting of variables, coefficients, and exponents. In this exercise, the polynomial is \( 5x^3 + 11x^3 + 2 \). Several characteristics of polynomials to remember include:

• Terms: Each part of a polynomial separated by a plus or minus sign, like \(5x^3\) and \(11x^3\).
• Coefficients: Numbers in front of the variable terms, such as 5 and 11 in our polynomial.
• Constants: Numbers without a variable, as seen here with the 2.

Polynomials can be simple or complex, and simplifying them often involves combining like terms and performing arithmetic operations. In our given exercise, the terms \( 5x^3 \) and \( 11x^3 \) are like terms and can be combined through addition before or after substitution.

Exponents

Exponents are a compact way to denote repeated multiplication of a number by itself. In our exercise, the exponent of 3 is shown in terms such as \( x^3 \). Key points about exponents include:

• An expression like \( x^3 \) is read as \( x \) to the power of 3, meaning \( x \cdot x \cdot x \).
• When substituting, calculate the power of the number first before any other operations to follow the order of operations correctly.
• The negative sign must be inside the parentheses for \( (-8)^3 \) to denote that \(-8\) is the base, making it \(-8 \times -8 \times -8\).

In our example, substituting \( -8 \) for \( x \) results in calculations of \( (-8)^3 \) which simplifies to \(-512\) for each term before applying the coefficients. Understanding and calculating exponents correctly can significantly impact the accuracy of solving polynomial equations.