Problem 84 - 1,001 GED Practice Questions For Dummies

Question

Simplify: \((5 x+1)(3 x-4)\) (A) \(15 x^2-17 x-4\) (B) \(-2 x-4\) (C) \(8 x-3\) (D) \(15 x^2+23 x-4\)

Step-by-Step Solution

Answer

(A) \(15x^2 - 17x - 4\)

Step 1: Multiply the first terms

Multiply the first terms of each binomial together: \( (5x) \times (3x) = 15x^2 \)

Step 2: Multiply the outer terms

Multiply the outer terms of each binomial together: \( (5x) \times (-4) = -20x \)

Step 3: Multiply the inner terms

Multiply the inner terms of each binomial together: \( (1) \times (3x) = 3x \)

Step 4: Multiply the last terms

Multiply the last terms of each binomial together: \( (1) \times (-4) = -4 \)

Step 5: Combine like terms

Combine the terms from Steps 2 and 3, as they are like terms: \( -20x + 3x = -17x \)

Step 6: Write the resulting expression

Combine the results from Steps 1, 5, and 4 to get the simplified expression: \( 15x^2 - 17x - 4 \) Comparing this expression with the given options, we find that the correct answer is (A) \(15x^2 - 17x - 4\).

Key Concepts

Binomial Multiplication

When you see expressions like \( (5x + 1)(3x - 4) \), it involves what's called binomial multiplication.A binomial is a polynomial with exactly two terms. Think of it like a two-word expression.To multiply two binomials, use the distributive property, sometimes remembered by the FOIL method, which stands for:

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

In this problem, we multiply:

• First: \( (5x) \times (3x) = 15x^2 \)
• Outer: \( (5x) \times (-4) = -20x \)
• Inner: \( (1) \times (3x) = 3x \)
• Last: \( (1) \times (-4) = -4 \)

This approach ensures each part of the binomial interacts with each other part, opening up the expression.

Combining Like Terms

After using the FOIL method for binomials, you often end up with terms that can be simplified.These are known as 'like terms.'Like terms have the same variable raised to the same power.For example, \(-20x\) and \(3x\) are like terms because both are terms in \(x\).
To combine them, just add or subtract the coefficients:

• Combine \(-20x + 3x \to -17x\).

Combining like terms is crucial because it simplifies the expression into its final form. Always check for like terms to keep polynomials as simple as possible.

Polynomials

Polynomials are expressions involving variables and coefficients. They can have one or more terms and are often used to represent various algebraic expressions.
In this exercise, the result of the binomial multiplication and combining like terms is a polynomial: \(15x^2 - 17x - 4\).

• \(15x^2\) is a term with \(x\) squared (a quadratic term).
• \(-17x\) is a linear term (involves \(x\) to the power of one).
• \(-4\) is a constant term (no variable attached).

Polynomials are organized by degrees, which is the highest power of the variable:

• In \(15x^2 - 17x - 4\), the degree is 2, making it a quadratic polynomial.

Understanding polynomials helps you recognize and predict behavior of expressions in various math problems.