Problem 5 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. $$ (5 x+2)(2 x+1)=?+9 x+2 $$

Step-by-Step Solution

Answer

The missing term in the right side of the equation is \(10x^2\). So the completed statement is: \((5x+2)(2x+1) = 10x^2 + 9x + 2\)

Step 1: Expand the Left Side

Expand \((5x + 2)(2x + 1)\) using the distributive property (FOIL method). When applied, it gives: \(10x^2 + 5x + 4x + 2 =10x^2 + 9x +2\)

Step 2: Compare with the Right Side

On comparing the obtained expression with the right side of the equation, which is \(?\) + \(9x\) + \(2\), it's evident that \(9x\) and \(2\) are present in both, but the left side also contains \(10x^2\) which is the missing term on the right side.

Key Concepts

Distributive Property

When dealing with polynomial expressions, the distributive property is a crucial concept. It essentially means that you distribute each term in one set of parentheses to each term in another. In simpler terms, you "multiply every term inside the parenthesis by the term outside." This is often used when you encounter expressions like

• \((a + b)(c + d)\).

You multiply "a" with both "c" and "d," then "b" with "c" and "d." This ensures that every combination of terms from both expressions is covered.

Applying the distributive property properly ensures that you account for each possible product, which may help reveal hidden terms or patterns in your mathematical work.

FOIL Method

The FOIL method is a specific application of the distributive property used to multiply two binomials. FOIL stands for First, Outer, Inner, and Last. These terms represent the sequence in which you multiply the terms of both binomials:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the innermost terms.
• Last: Multiply the last terms of each binomial.

For example, for \((5x + 2)(2x + 1)\),

• First: \(5x \cdot 2x = 10x^2\)
• Outer: \(5x \cdot 1 = 5x\)
• Inner: \(2 \cdot 2x = 4x\)
• Last: \(2 \cdot 1 = 2\)

After performing all these multiplications, you combine the like terms to simplify the expression, resulting in \(10x^2 + 9x + 2\).

Mastering the FOIL method helps make working with polynomials clear and manageable.

Quadratic Expression

A quadratic expression is a polynomial of degree two. It generally takes the form \(ax^2 + bx + c\),where \(a\), \(b\), and \(c\) are constants.

Quadratics appear frequently in math due to their wide application in various problem-solving scenarios. For example, in motions, physics, and economics. The expression \(10x^2 + 9x + 2\)that results from expanding and simplifying \((5x + 2)(2x + 1)\) is a classic example. Here,

• \(a = 10\)
• \(b = 9\)
• \(c = 2\).

Identifying these terms in a quadratic helps you apply different methods of solving, like factoring or using the quadratic formula.

Understanding how to work with quadratics is critical for succeeding in many areas of math and science.