Problem 5 - Algebra 1: Concepts and Skills

Question

Copy the equation and fill in the blanks. \((x-3)(x+1)=x^{2}-2 x-\)______

Step-by-Step Solution

Answer

The constant term is -3

Step 1: Distributive Property (First)

Multiply the first terms in each parentheses, which are x and x. This gives: \(x*x=x^{2}\)

Step 2: Distributive Property (Outside)

Multiply the outside terms of the parentheses, which are x and 1. Then add the obtained value to the result from Step 1. This gives us: \(x^{2} + x(1) = x^{2} + x\)

Step 3: Distributive Property (Inside)

Next, multiply the inside terms, which are -3 and x. Add this to the previous steps, resulting in: \(x^{2} + x -3x = x^{2} -2x\)

Step 4: Distributive Property (Last)

Finally, multiply the last terms in each parentheses, which are -3 and 1. Adding this to the rest gives: \(x^{2} -2x -3=-1\)

Step 5: Result

From these steps we see that the constant term of the expanded equation is -3

Key Concepts

Distributive Property

The distributive property is a fundamental principle used in algebra, simplifying expressions, and solving equations. It states that for all real numbers a, b, and c, the expression \( a(b + c) \) is equivalent to \( ab + ac \). In the context of quadratic equations, this property is essential when multiplying binomials. When dealing with the expression \((x-3)(x+1)\), we apply the distributive property to ensure each term in the first parenthesis is multiplied by every term in the second parenthesis.

• First, multiply the first terms: \(x \times x = x^2\)
• Second, multiply the outer terms: \(x \times 1 = x\)
• Then, multiply the inner terms: \((-3) \times x = -3x\)
• Finally, multiply the last terms: \((-3) \times 1 = -3\)

These calculations show the step-by-step application of the distributive property to achieve each part of the quadratic expression.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together to form a single polynomial. This operation often includes multiple uses of the distributive property. Let's consider the multiplication of two binomials, \((x-3)\) and \((x+1)\), used in the given exercise.The FOIL method is a common approach for multiplying two binomials:

• First: Multiply the first terms in each binomial: \(x \times x = x^2\)
• Outside: Multiply the outer terms: \(x \times 1 = x\)
• Inside: Multiply the inner terms: \(-3 \times x = -3x\)
• Last: Multiply the last terms: \(-3 \times 1 = -3\)

Adding these results gives us the polynomial \(x^2 + x - 3x - 3\). Simplifying by combining like terms, we obtain \(x^2 - 2x - 3\). This final expression achieves the goal of combining all terms into a single polynomial.

Factoring Quadratics

Factoring quadratics is the process of expressing a quadratic equation as a product of two binomial expressions. This is the reverse operation of expanding binomials. Consider the equation \(x^2 - 2x - 3\), which we obtained from expanding the binomials \((x-3)(x+1)\).To factor this quadratic, look for two numbers whose product equals the constant term (-3) and whose sum equals the linear coefficient (-2). In this case, the numbers are -3 and 1.

• The product of \(-3 \times 1 = -3\)
• The sum of \(-3 + 1 = -2\)

Thus, the quadratic \(x^2 - 2x - 3\) factors back into \((x-3)(x+1)\). Reversing the expansion process maintains the original structure, which can be useful for solving the equation or identifying its roots.