Problem 9
Question
For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(-3 x^{2}+6 x+24\right) \div(x-4) $$
Step-by-Step Solution
Verified Answer
The remainder is 0.
1Step 1: Understand the Remainder Theorem
The Remainder Theorem states that for a polynomial \( f(x) \) divided by \( (x - c) \), the remainder is \( f(c) \). This means we just need to evaluate the polynomial at \( x = c \) to find the remainder.
2Step 2: Identify \( c \) in the Divisor
In the expression \((x-4)\), the \( c \) value is \( 4 \). This is the value that we will substitute into the polynomial \(-3x^2 + 6x + 24\) to find the remainder.
3Step 3: Evaluate the Polynomial at \( x = 4 \)
Substitute \( x = 4 \) into the polynomial \(-3x^2 + 6x + 24\). Perform the substitution: \[ f(4) = -3(4)^2 + 6(4) + 24 \] Calculate each term:\( -3(4)^2 = -3 \times 16 = -48 \)\( 6(4) = 24 \)Sum the values:\( -48 + 24 + 24 = 0 \) Thus, the remainder is 0.
Key Concepts
Polynomial DivisionEvaluating PolynomialsAlgebraic Expressions
Polynomial Division
Polynomial division is a process used to divide one polynomial by another polynomial. Similar to long division in arithmetic, polynomial division helps simplify expressions and find specifics, like the quotient and remainder.
When we perform polynomial division, we are often finding out how many times one polynomial "fits" into another. In the specific exercise you have, the polynomial \(-3x^2 + 6x + 24\) is divided by \((x - 4)\). Here, \(x - 4\) is a linear polynomial, making the division a bit simpler than dealing with higher degrees.
The key outcome of this type of division is to determine the remainder, which tells you what remains after the polynomial \(-3x^2 + 6x + 24\) has been divided by \(x - 4\). This is crucial for checking how well the divisor fits into the original polynomial.
When we perform polynomial division, we are often finding out how many times one polynomial "fits" into another. In the specific exercise you have, the polynomial \(-3x^2 + 6x + 24\) is divided by \((x - 4)\). Here, \(x - 4\) is a linear polynomial, making the division a bit simpler than dealing with higher degrees.
The key outcome of this type of division is to determine the remainder, which tells you what remains after the polynomial \(-3x^2 + 6x + 24\) has been divided by \(x - 4\). This is crucial for checking how well the divisor fits into the original polynomial.
Evaluating Polynomials
Evaluating polynomials involves substituting a specific value for the variable in a polynomial expression and then performing the arithmetic to find the result.
In the context of the Remainder Theorem, you evaluate a polynomial at a given value to find the remainder of the division of the polynomial by a binomial like \((x-c)\).
In our example, we evaluated the polynomial function \(-3x^2 + 6x + 24\) at \(x = 4\) using the formula \(f(4)\). Here's how:
In the context of the Remainder Theorem, you evaluate a polynomial at a given value to find the remainder of the division of the polynomial by a binomial like \((x-c)\).
In our example, we evaluated the polynomial function \(-3x^2 + 6x + 24\) at \(x = 4\) using the formula \(f(4)\). Here's how:
- Substituted 4 in place of every \(x\) in the polynomial.
- Calculated each term: \(-3(4)^2 = -48\) and \(6(4) = 24\).
- Summed up the values: \(-48 + 24 + 24 = 0\).
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols such as addition, subtraction, multiplication, and division. Understanding how to manipulate and simplify these expressions is fundamental in algebra.
When working with algebraic expressions, it is essential to understand different forms and operations, such as factorization and expansion. In your exercise, the expression \(-3x^2 + 6x + 24\) is considered an algebraic expression.
This specific expression can be broken down into terms, which are typically separated by plus or minus signs.
When working with algebraic expressions, it is essential to understand different forms and operations, such as factorization and expansion. In your exercise, the expression \(-3x^2 + 6x + 24\) is considered an algebraic expression.
This specific expression can be broken down into terms, which are typically separated by plus or minus signs.
- \(-3x^2\) is one term which is quadratic.
- \(6x\) is another term which is linear.
- \(24\) is a constant term.
Other exercises in this chapter
Problem 9
For the following exercises, find the inverse of the function on the given domain. $$ f(x)=12-x^{2},[0, \infty) $$
View solution Problem 9
For the following exercises, find the domain of the rational functions. $$ f(x)=\frac{x^{2}+4 x-3}{x^{4}-5 x^{2}+4} $$
View solution Problem 9
For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{2}-3 x+2\right) \div(x+2) $$
View solution Problem 9
For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ C(t)=2 t(t-3)(t+1)^{2} $$
View solution