Find the Degree, Leading Term, and Leading Coefficient g(n)=-(3n-1)(2n+1)
The given problem involves determining three key aspects of a polynomial function called g(n). The task is to find:
Degree: This is the highest power of the variable (in this case, 'n') within the polynomial.
Leading Term: This is the term in the polynomial that contains the highest power of the variable.
Leading Coefficient: This is the coefficient of the leading term, essentially the numerical factor attached to the term with the highest degree.
The polynomial g(n) is given in factored form as the product of two binomials: -(3n-1) and (2n+1). To find the requested attributes, one would typically first expand the polynomial (multiply the binomials), then identify the term with the highest power of 'n' and extract the relevant information from it.
$g \left(\right. n \left.\right) = - \left(\right. 3 n - 1 \left.\right) \left(\right. 2 n + 1 \left.\right)$
Answer
Solution:
Step 1: Expand and simplify the given polynomial expression.
Step 1.1: Perform the multiplication of the binomials.
Step 1.1.1: Use the distributive property to set up the multiplication.
$$ g(n) = -(3n - 1)(2n + 1) $$
Step 1.1.2: Begin the multiplication process.
Step 1.1.2.1: Multiply the first terms.
$$ g(n) = -(3n)(2n) $$
Step 1.1.2.2: Multiply the outer, inner, and last terms.
$$ g(n) = -(3n)(2n) - (3n)(1) + (1)(2n) + (1)(1) $$
Step 1.2: Apply the FOIL method to expand the binomials.
Step 1.2.1: Distribute each term in the first binomial across the second binomial.
$$ g(n) = -3n(2n + 1) + 1(2n + 1) $$
Step 1.2.2: Continue distributing to expand fully.
$$ g(n) = -3n(2n) - 3n(1) + 1(2n) + 1(1) $$
Step 1.3: Combine like terms and simplify the expression.
Step 1.3.1: Simplify each term individually.
Step 1.3.1.1: Use the commutative property to rearrange multiplication.
$$ g(n) = -3 \cdot (2n^2) - 3n + 2n + 1 $$
Step 1.3.1.2: Add exponents when multiplying powers of the same base.
Step 1.3.1.2.1: Combine the n terms.
$$ g(n) = -3 \cdot (2n^2) - 3n + 2n + 1 $$
Step 1.3.1.2.2: Perform the multiplication of constants and variables.
$$ g(n) = -6n^2 - 3n + 2n + 1 $$
Step 1.3.1.3: Multiply the coefficients.
$$ g(n) = -6n^2 - 3n + 2n + 1 $$
Step 1.3.1.4: Continue simplifying by combining like terms.
$$ g(n) = -6n^2 - n + 1 $$
Step 1.3.2: Final simplification.
$$ g(n) = -6n^2 - n + 1 $$
Step 2: Determine the degree of the polynomial.
Step 2.1: Identify the degree of each term.
$$ -6n^2 \rightarrow 2, \quad -n \rightarrow 1, \quad 1 \rightarrow 0 $$
Step 2.2: The highest degree is the degree of the polynomial.
Degree: $2$
Step 3: Identify the leading term of the polynomial.
Leading Term: $-6n^2$
Step 4: Find the leading coefficient of the polynomial.
Step 4.1: The leading term is the term with the highest power of n.
Leading Term: $-6n^2$
Step 4.2: The coefficient of the leading term is the leading coefficient.
Leading Coefficient: $-6$
Step 5: Compile the results.
Polynomial Degree: $2$ Leading Term: $-6n^2$ Leading Coefficient: $-6$
Knowledge Notes:
Polynomial: A mathematical expression consisting of variables (also known as indeterminates), coefficients, and non-negative integer exponents of variables. Polynomials are summed together and can include constants.
Degree of a Polynomial: The highest degree of any term in the polynomial. The degree of a term is the sum of the exponents of the variables that appear in it, and it reflects the highest power of the variable in the polynomial.
Leading Term: In a polynomial, the leading term is the term with the highest degree. When the polynomial is written in standard form (descending powers of the variable), the leading term is the first term.
Leading Coefficient: The coefficient of the leading term in a polynomial.
Distributive Property: A property that allows us to multiply a sum by multiplying each addend separately and then sum the products. It is often used in the expansion of binomials.
FOIL Method: A technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the terms that are multiplied together.
Commutative Property of Multiplication: A property stating that the order in which two numbers are multiplied does not affect the product.
Combining Like Terms: A process in algebra where terms with the same variables and exponents are summed together to simplify an expression.
