Problem 19
Question
For the following problems, a term will be given followed by a group of its factors. List the coefficient of the given group of factors. $$ 7 y(y+3), 7 y $$
Step-by-Step Solution
Verified Answer
Answer: The coefficient of \(7y\) in the term \(7y(y+3)\) is \((y+3)\).
1Step 1: Identify the term and the group of factors
The given term is \(7y(y+3)\) and the group of factors we want the coefficient for is \(7y\).
2Step 2: Remove the common factors between the term and the group of factors
We need to divide the given term by the group of factors that we want the coefficient for. So, divide \(7y(y+3)\) by \(7y\) to get the coefficient: \(\frac{7y(y+3)}{7y}\).
3Step 3: Perform the division
Now, perform the division: \(\frac{7y(y+3)}{7y}\) = \((y+3)\).
4Step 4: Identify the coefficient of the given group of factors
Since the result of the division is \((y+3)\), that means the coefficient of \(7y\) in the original term \(7y(y+3)\) is \((y+3)\).
Key Concepts
CoefficientsFactorsPolynomials
Coefficients
In algebra, a coefficient is a number or a constant that is multiplied by a variable in a term. Imagine the coefficient as the numerical part of a term that gives it its magnitude. For example, in the term \(7y(x + 2)\), the number \(7\) is the coefficient of \(yx\) or \(7y\).
Coefficients tell us "how many" of the variable we have and are crucial when simplifying or evaluating algebraic expressions.
In the context of the exercise, finding the coefficient involves dividing the entire term by the specified group of factors to see what is left, which directly corresponds to the coefficient of these factors.
Coefficients tell us "how many" of the variable we have and are crucial when simplifying or evaluating algebraic expressions.
In the context of the exercise, finding the coefficient involves dividing the entire term by the specified group of factors to see what is left, which directly corresponds to the coefficient of these factors.
Factors
Factors in algebra are expressions that can be multiplied together to get another expression. Simply put, a factor of a term is something that divides it exactly, leaving no remainder.
For the expression \(7y(y+3)\), both \(7y\) and \((y+3)\) are factors, since multiplying them gives you the original term.
When you look for a group of factors in a term, you're essentially identifying elements of the term that work together multiplicatively. In the provided exercise, the group of factors being considered was \(7y\), and knowing how to isolate such factors from the rest can help simplify complex algebraic equations.
For the expression \(7y(y+3)\), both \(7y\) and \((y+3)\) are factors, since multiplying them gives you the original term.
When you look for a group of factors in a term, you're essentially identifying elements of the term that work together multiplicatively. In the provided exercise, the group of factors being considered was \(7y\), and knowing how to isolate such factors from the rest can help simplify complex algebraic equations.
Polynomials
Polynomials are expressions made up of variables and coefficients, involving terms with non-negative integer exponents like \(x^2\), \(3x\), or \(7xy\). A polynomial can have multiple terms, and an important aspect is that each term consists of numbers (coefficients) and variables raised to powers.
In the problem given, \(7y(y+3)\) represents a polynomial with two terms fundamentally included within the parenthesis. Even when dealing with seemingly straightforward expressions, understanding them as parts of a more complex polynomial can offer insights when simplifying or expanding the expression.
Dividing polynomials can often reveal simpler forms, like observing the effect of removing a factor, similar to the division performed in the exercise.
In the problem given, \(7y(y+3)\) represents a polynomial with two terms fundamentally included within the parenthesis. Even when dealing with seemingly straightforward expressions, understanding them as parts of a more complex polynomial can offer insights when simplifying or expanding the expression.
Dividing polynomials can often reveal simpler forms, like observing the effect of removing a factor, similar to the division performed in the exercise.
Other exercises in this chapter
Problem 19
Use numerical evaluation on the equations. Geometry (area of a rectangle) \(A=l w . \quad\) Find \(A\) if \(l=15\) and \(w=9\).
View solution Problem 19
Simplify the algebraic expressions for the following problems. $$ 6 a(a-2)-\left(2 a^{2}+a-11\right) $$
View solution Problem 20
Name the property of real numbers that makes \(4 y x^{2}=4 x^{2} y\) a true statement.
View solution Problem 20
For the following problems, find the products. $$ (a-10)^{2} $$
View solution