Problem 235
Question
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{3}+x^{2}-20 x $$
Step-by-Step Solution
Verified Answer
The \(x\)-intercepts are \( x = 0 \), \( x = -5 \), and \( x = 4 \).
1Step 1: Set the function equal to zero
To find the intercepts, set the function equal to zero: \[ f(x) = x^3 + x^2 - 20x = 0 \]
2Step 2: Factor the polynomial
Find common factors of the terms. Each term in \( f(x) \) has an \( x \), so factor out \( x \):\[ x(x^2 + x - 20) = 0 \]
3Step 3: Factor the quadratic expression
Focus on factoring the quadratic expression inside the parenthesis: \( x^2 + x - 20 \). We need two numbers that multiply to -20 and add to 1. These numbers are 5 and -4:\[ x^2 + x - 20 = (x + 5)(x - 4) \]
4Step 4: Find all factors
Plug the factors of the quadratic expression into the factored form:\[ x(x + 5)(x - 4) = 0 \]
5Step 5: Apply the Zero Product Property
Set each factor in the product equal to zero: 1. \( x = 0 \) 2. \( x + 5 = 0 \) implies \( x = -5 \) 3. \( x - 4 = 0 \) implies \( x = 4 \)
6Step 6: List the x-intercepts
The solutions from the previous step are the \( x \)-intercepts of the function. Therefore, the \( x \)-intercepts are \( x = 0 \), \( x = -5 \), and \( x = 4 \).
Key Concepts
x-interceptsfactoring polynomialsZero Product Propertyquadratic expressions
x-intercepts
An x-intercept is a point where a graph crosses the x-axis. At this point, the value of the function is equal to zero. This means that if you input the x-coordinate of the intercept into the function, the output will be zero.
To find x-intercepts for a polynomial function like \(f(x)=x^3+x^2-20x\), set the entire function equal to zero and solve for x. This process is essential because it highlights where the function hits the x-axis, giving us insight into the behavior of the graph.
In our specific example, by solving \(x(x+5)(x-4)=0\), we find the x-intercepts at \(x=0\), \(x=-5\), and \(x=4\). Each intercept corresponds to a real number root of the equation which intersects the x-axis.
To find x-intercepts for a polynomial function like \(f(x)=x^3+x^2-20x\), set the entire function equal to zero and solve for x. This process is essential because it highlights where the function hits the x-axis, giving us insight into the behavior of the graph.
In our specific example, by solving \(x(x+5)(x-4)=0\), we find the x-intercepts at \(x=0\), \(x=-5\), and \(x=4\). Each intercept corresponds to a real number root of the equation which intersects the x-axis.
factoring polynomials
Factoring polynomials is an essential algebraic skill used to simplify and solve polynomial equations. It involves breaking down a polynomial into simpler polynomials, known as factors, that multiply together to give the original polynomial.
When faced with a polynomial like \(x^3 + x^2 - 20x\), the first step is often to look for common factors that every term shares. Here, each term has an \(x\), allowing us to factor it out:
When faced with a polynomial like \(x^3 + x^2 - 20x\), the first step is often to look for common factors that every term shares. Here, each term has an \(x\), allowing us to factor it out:
- Common Factor: Extract \(x\) to get \(x(x^2 + x - 20)\).
- Quadratic Factor: \(x^2 + x - 20\) becomes \((x+5)(x-4)\) by identifying numbers that multiply to -20 and add to 1.
Zero Product Property
The Zero Product Property is a powerful concept in algebra that states if the product of multiple factors equals zero, then at least one of the factors must be zero.
Applying this property helps solve polynomial equations efficiently. Once a polynomial is factored, each factor is set to zero to solve for the possible values of the variable. For the polynomial \(x(x+5)(x-4)=0\), set each factor equal to zero:
Applying this property helps solve polynomial equations efficiently. Once a polynomial is factored, each factor is set to zero to solve for the possible values of the variable. For the polynomial \(x(x+5)(x-4)=0\), set each factor equal to zero:
- \(x=0\)
- \(x+5=0\) leads to \(x=-5\)
- \(x-4=0\) leads to \(x=4\)
quadratic expressions
Quadratic expressions are polynomials of degree two, typically written in the form \(ax^2+bx+c\). These expressions are common in many algebraic contexts and are crucial for solving polynomial equations.
When factoring a quadratic, like \(x^2+x-20\), identify two numbers that multiply to the constant term (in this case, -20) and add up to the linear coefficient (here, 1). For this expression, the numbers are 5 and -4. Thus, the quadratic factors as \((x+5)(x-4)\).
Decomposing quadratics into a product of binomials simplifies solving for x-intercepts. The process requires:
When factoring a quadratic, like \(x^2+x-20\), identify two numbers that multiply to the constant term (in this case, -20) and add up to the linear coefficient (here, 1). For this expression, the numbers are 5 and -4. Thus, the quadratic factors as \((x+5)(x-4)\).
Decomposing quadratics into a product of binomials simplifies solving for x-intercepts. The process requires:
- Finding two numbers that multiply to \(c\) and add to \(b\).
- Writing the quadratic as a product of two binomial expressions.
Other exercises in this chapter
Problem 233
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=4 t^{4}+12 t^{3}-40 t^{2} $$
View solution Problem 234
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{4}-x^{2} $$
View solution Problem 236
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{3}+6 x^{2}-7 x $$
View solution Problem 237
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{3}+x^{2}-4 x-4 $$
View solution