Problem 6 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(x^{2}+x-56=(x \square)(x \square\))

Step-by-Step Solution

Answer

The factorized form is \((x + 8)(x - 7)\).

Step 1: Understanding the Problem

The exercise is asking us to factor the quadratic expression \(x^2 + x - 56\). Factoring means expressing it as a product of two binomials. The generic form for factoring a quadratic \(ax^2 + bx + c\) is \((x + p)(x + q)\) where \(p\) and \(q\) are numbers that add up to \(b\) (the coefficient of \(x\)) and multiply together to \(c\) (the constant term).

Step 2: Identifying p and q

We need to find two numbers \(p\) and \(q\) such that:1. \(p \cdot q = -56\) (the constant term)2. \(p + q = 1\) (the coefficient of \(x\))

Step 3: Testing Possible Values for p and q

Consider the pairs of factors of -56 and determine whether their sum is 1:- \(7\) and \(-8\): \(7 - 8 = -1\) (does not work)- \(8\) and \(-7\): \(8 - 7 = 1\) (works)Thus, \(p = 8\) and \(q = -7\).

Step 4: Writing the Factorized Form

Now that we have found \(p\) and \(q\), we can write the factorized form of the quadratic expression as:\((x + 8)(x - 7)\).

Step 5: Verifying the Factorization

To ensure the factorization is correct, expand \((x + 8)(x - 7)\) to check if it matches the original expression:- First, use the distributive property: \((x)(x) + (x)(-7) + (8)(x) + (8)(-7) = x^2 - 7x + 8x - 56 = x^2 + x - 56\).- The expanded form \(x^2 + x - 56\) matches the original, confirming the factorization is correct.

Key Concepts

Binomials

A binomial is simply a polynomial with two terms. In the context of factoring, a binomial typically takes the form \(x + p\) or \(x + q\), where \(p\) and \(q\) are constants. These terms \(p\) and \(q\) play crucial roles when breaking down a quadratic equation into a product of two binomials.
In this exercise, we factored the quadratic \(x^2 + x - 56\) into the product \((x + 8)(x - 7)\). Here, each of these expressions, \((x + 8)\) and \((x - 7)\), represents a binomial.
When working with binomials, it's important to remember that the sum and the product of the constant terms \(p\) and \(q\) must satisfy certain conditions to equate back to the original quadratic coefficients.

• The sum of \(p\) and \(q\) must equal the coefficient of the middle term (which is \(+1\) in \(x^2 + x - 56\)).
• The product of \(p\) and \(q\) must equal the constant term (which is \(-56\)).

Factoring quadratics into binomials is like piecing together a puzzle, where both pieces must fit perfectly to recreate the original expression.

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, meaning the highest power of the variable is squared. These typically take the general form \(ax^2 + bx + c\).
In our specific problem, we have the quadratic expression \(x^2 + x - 56\). This can be rewritten in the general form where \(a = 1\), \(b = 1\), and \(c = -56\).
The goal of factoring a quadratic expression is to express it as the product of two binomials, transforming it from a second-degree polynomial into a product of first-degree polynomials. This process reverses multiplication, splitting the expression into simpler parts.
Applying this to \(x^2 + x - 56\), we seek numbers that multiply to give \(-56\) (the constant \(c\)) and add up to \(1\) (the coefficient \(b\)). Solving this step allows us to break down the quadratic expression into: \((x + 8)(x - 7)\). This expression remains equal to the original when expanded, validating the factorization process.

Distributive Property

The distributive property is a fundamental principle used in algebra that allows you to multiply a single term by two or more terms inside a parenthesis. It states that \((a(b + c) = ab + ac)\).
This property is essential when factoring or expanding expressions, such as converting \((x + 8)(x - 7)\) back to \(x^2 + x - 56\). To expand, we apply the distributive property as follows:

• First, multiply \(x\) by each term in \((x - 7)\): \((x)(x) + (x)(-7)\) gives \((x^2 - 7x)\).
• Then, multiply \(+8\) by each term in \((x - 7)\): \((8)(x) + (8)(-7)\) gives \((8x - 56)\).

Finally, combine these results: \(x^2 - 7x + 8x - 56\). Simplifying by combining like terms (\

Quadratic Expressions

Quadratic expressions are polynomials of degree 2, meaning the highest power of the variable is squared. These typically take the general form \(ax^2 + bx + c\).
In our specific problem, we have the quadratic expression \(x^2 + x - 56\). This can be rewritten in the general form where \(a = 1\), \(b = 1\), and \(c = -56\).
The goal of factoring a quadratic expression is to express it as the product of two binomials, transforming it from a second-degree polynomial into a product of first-degree polynomials. This process reverses multiplication, splitting the expression into simpler parts.
Applying this to \(x^2 + x - 56\), we seek numbers that multiply to give \(-56\) (the constant \(c\)) and add up to \(1\) (the coefficient \(b\)). Solving this step allows us to break down the quadratic expression into: \((x + 8)(x - 7)\). This expression remains equal to the original when expanded, validating the factorization process.

Distributive Property

The distributive property is a fundamental principle used in algebra that allows you to multiply a single term by two or more terms inside a parenthesis. It states that \((a(b + c) = ab + ac)\).
This property is essential when factoring or expanding expressions, such as converting \((x + 8)(x - 7)\) back to \(x^2 + x - 56\). To expand, we apply the distributive property as follows:

• First, multiply \(x\) by each term in \((x - 7)\): \((x)(x) + (x)(-7)\) gives \((x^2 - 7x)\).
• Then, multiply \(+8\) by each term in \((x - 7)\): \((8)(x) + (8)(-7)\) gives \((8x - 56)\).

Finally, combine these results: \(x^2 - 7x + 8x - 56\). Simplifying by combining like terms \(( -7x + 8x = x)\) results in the final expanded expression \(x^2 + x - 56\), proving the factorization and distributive process are correct.