Simplify (7m^2+42m-49)/(m+7)
The given problem is asking for the simplification of a rational algebraic expression, which is a fraction where the numerator and the denominator are polynomials. Specifically, the expression to be simplified is (7m^2+42m-49)/(m+7). The task involves reducing the expression to its simplest form by factoring the numerator, the polynomial 7m^2+42m-49, and then canceling out any common factors that appear in both the numerator and the denominator. The denominator in this case is a linear polynomial (m+7). Factors of the numerator that match the denominator can be simplified, resulting in a more streamlined algebraic expression. This process is commonly known in algebra as simplifying fractions or reducing fractions to their lowest terms.
$\frac{7 m^{2} + 42 m - 49}{m + 7}$
Answer
Solution:
Step 1: Simplify the Numerator
Step 1.1: Extract the Common Factor
Extract the common factor of 7 from the numerator expression $7m^2 + 42m - 49$.
Step 1.1.1: Factor Out from $7m^2$
Extract 7 from $7m^2$: $\frac{7(m^2) + 42m - 49}{m + 7}$
Step 1.1.2: Factor Out from $42m$
Extract 7 from $42m$: $\frac{7(m^2) + 7(6m) - 49}{m + 7}$
Step 1.1.3: Factor Out from $-49$
Extract 7 from $-49$: $\frac{7m^2 + 7(6m) + 7(-7)}{m + 7}$
Step 1.1.4: Combine Factored Terms
Combine the terms factored by 7: $\frac{7(m^2 + 6m - 7)}{m + 7}$
Step 1.2: Factor the Quadratic Expression
Factor the quadratic $m^2 + 6m - 7$ by finding two numbers that multiply to $-7$ and add to $6$.
Step 1.2.1: Apply the AC Method
Identify two numbers with a product of $ac$ (where $a$ is the coefficient of $m^2$ and $c$ is the constant term) and a sum of $b$ (the coefficient of $m$): $-1$ and $7$.
Step 1.2.2: Write the Factored Form
Write the numerator in its factored form: $\frac{7((m - 1)(m + 7))}{m + 7}$
Step 2: Simplify Terms
Step 2.1: Cancel Common Factors
Cancel out the common factor of $(m + 7)$ in the numerator and denominator.
Step 2.1.1: Perform the Cancellation
Perform the cancellation: $\frac{7(m - 1)\cancel{(m + 7)}}{\cancel{m + 7}}$
Step 2.1.2: Simplify the Expression
Simplify the expression to $7(m - 1)$
Step 2.2: Distribute the 7
Apply the distributive property to $7(m - 1)$.
Step 2.3: Final Simplification
Multiply 7 by each term in the parentheses: $7m - 7$
Knowledge Notes:
Common Factor: In algebra, factoring out the greatest common factor (GCF) is a method used to simplify expressions. The GCF is the largest factor that divides all terms in the expression.
Quadratic Factoring: Factoring a quadratic expression involves rewriting it as a product of two binomials. The AC method is one approach to factoring quadratics, where you look for two numbers that multiply to the product of the coefficient of $x^2$ and the constant term (ac), and add up to the coefficient of $x$ (b).
Cancellation Law: In fractions, if a factor appears in both the numerator and the denominator, it can be cancelled out. This is based on the property that any number divided by itself equals one.
Distributive Property: This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. It is often used in algebra to simplify expressions and is expressed as $a(b + c) = ab + ac$.
Simplification: The process of reducing an expression to its simplest form. This often involves factoring, expanding, and cancelling out terms where possible.
In the given problem, the simplification process involves factoring out the GCF, using the AC method to factor the quadratic, cancelling common factors, and applying the distributive property to reach the final simplified form of the expression.
