Problem 20
Question
Either guess a zero by observation or use a graphing calculator to guess a root; then use long division. $$ g(x)=3 x^{3}+3 $$
Step-by-Step Solution
Verified Answer
The root of the given function \(g(x)=3x^{3}+3\) is \(x=-1\). After dividing \(g(x)\) by its corresponding factor \((x+1)\), the quotient function is \(3x^{2}-3x+3\), and hence, \(g(x)=(3x^{2}-3x+3)(x+1)\).
1Step 1: Identifying a Possible Root
Through observation, it can be guessed that \(x=-1\) is a root of the function \(g(x)=3x^{3}+3\). This is because when we substitute \(x=-1\) into the function, it satisfies the condition \(g(x)=0\).
2Step 2: Confirming the Root by Long Division
Now, we need to confirm if \(x=-1\) is a root, and if so, what the resulting function is if we divide \(g(x)\) by \((x+1)\) (because the root is negative, the factor will be positive). This can be done through long division: Divide \(3x^{3}+3\) by \(x+1\). We get \(3x^{2}-3x+3\), with a remainder of 0. The remainder being 0 confirms that \(x=-1\) is indeed a root.
3Step 3: Summary of the Result
So, the root of the function was correctly identified as \(x=-1\) and the resulting function after division by its corresponding factor \((x+1)\) is \(3x^{2}-3x+3\). Thus, the original function can be expressed as \(g(x)=(3x^{2}-3x+3)(x+1)\).
Key Concepts
Roots of PolynomialsLong Division MethodGraphing Calculators
Roots of Polynomials
Finding the roots of polynomial functions is a central aspect of algebra and pre-calculus. A "root" of a polynomial is a solution to the equation when the polynomial is set equal to zero. Essentially, it's a value that makes the polynomial equal zero when substituted for the variable. Consider the polynomial \( g(x) = 3x^3 + 3 \). To determine its roots, one common approach is to check possible values by substituting them into the polynomial.
In our example, by inspection or using a graphing calculator, we might identify \( x = -1 \) as a candidate for a root. Verification requires substituting \( x = -1 \) into the polynomial. If \( g(-1) = 0 \), then \( -1 \) is indeed a root. This approach is sometimes called "guess and check," which can be effective for simpler polynomials or when aided by technology.
In our example, by inspection or using a graphing calculator, we might identify \( x = -1 \) as a candidate for a root. Verification requires substituting \( x = -1 \) into the polynomial. If \( g(-1) = 0 \), then \( -1 \) is indeed a root. This approach is sometimes called "guess and check," which can be effective for simpler polynomials or when aided by technology.
Long Division Method
The long division method allows you to divide polynomials in a way similar to numeric long division. This technique is useful for confirming a suspected root and simplifying the polynomial by factoring it.
In the context of our polynomial \( g(x) = 3x^3 + 3 \), we suspect \( x = -1 \) is a root. We convert this root to a factor, \( (x + 1) \), and perform polynomial long division to divide \( 3x^3 + 3 \) by \( x + 1 \).
Steps for polynomial long division include:
In the context of our polynomial \( g(x) = 3x^3 + 3 \), we suspect \( x = -1 \) is a root. We convert this root to a factor, \( (x + 1) \), and perform polynomial long division to divide \( 3x^3 + 3 \) by \( x + 1 \).
Steps for polynomial long division include:
- Divide the first term of the dividend by the first term of the divisor.
- Multiply the entire divisor by the result and subtract from the dividend.
- Repeat with the new (reduced) dividend until all terms are divided.
Graphing Calculators
Graphing calculators are invaluable tools when dealing with polynomials, particularly for visualizing roots. They allow students to quickly and accurately plot graphs of polynomial functions. The visual output helps identify where the function crosses the x-axis, indicating potential roots.
Utilizing a graphing calculator with \( g(x) = 3x^3 + 3 \), you would plot the graph and observe where the curve intersects the x-axis. This intersection represents the root. Graphing calculators often include functions for calculating zeroes of a function, offering precise values and verifying roots guessed by observation.
With the combination of graphing abilities and computational tools, these calculators make exploring complex polynomials less daunting. They facilitate confirming roots suggested by guess and check or algebraic methods, making them essential for students to master polynomial concepts efficiently.
Utilizing a graphing calculator with \( g(x) = 3x^3 + 3 \), you would plot the graph and observe where the curve intersects the x-axis. This intersection represents the root. Graphing calculators often include functions for calculating zeroes of a function, offering precise values and verifying roots guessed by observation.
With the combination of graphing abilities and computational tools, these calculators make exploring complex polynomials less daunting. They facilitate confirming roots suggested by guess and check or algebraic methods, making them essential for students to master polynomial concepts efficiently.
Other exercises in this chapter
Problem 19
Either guess a zero by observation or use a graphing calculator to guess a root; then use long division. $$ P(x)=x^{4}-2 x^{3}-6 x^{2}+12 x $$
View solution Problem 19
Graph the function given, labeling all \(x\) -intercepts, \(y\) intercepts, and the \(x\) - and \(y\) -coordinates of any local maximum and minimum points. $$ f
View solution Problem 20
Graph the function given, labeling all \(x\) -intercepts, \(y\) intercepts, and the \(x\) - and \(y\) -coordinates of any local maximum and minimum points. $$ f
View solution Problem 21
Either guess a zero by observation or use a graphing calculator to guess a root; then use long division. $$ h(x)=x^{3}-8 $$
View solution