Problem 15
Question
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ f(x)=x^{3}+x^{2}-4 x-4 $$
Step-by-Step Solution
Verified Answer
The x-intercepts are at \((-1, 0)\), \((2, 0)\), and \((-2, 0)\).
1Step 1: Set the Function Equal to Zero
To find the intercepts of the polynomial function, set the equation:\[ f(x)=x^{3}+x^{2}-4x-4 = 0 \]
2Step 2: Try Rational Root Theorem
According to the Rational Root Theorem, the possible rational roots are the factors of the constant term (-4), divided by the factors of the leading coefficient (1). Thus, the possible rational roots are \(\pm 1, \pm 2, \pm 4\).
3Step 3: Test Possible Rational Roots
Substitute each potential root into the polynomial until you find one that zeroes the function:- For \(x = 1\), \((1)^3 + (1)^2 - 4(1) - 4 = -6\)- For \(x = -1\), \((-1)^3 + (-1)^2 - 4(-1) - 4 = 0\)Thus, \(x = -1\) is a root.
4Step 4: Factor the Polynomial
Use the found root \(x = -1\) to factor the polynomial by polynomial division or synthetic division against \(x + 1\). Divide \(x^{3}+x^{2}-4x-4\) by \(x + 1\). The quotient will be \(x^2 - 4\).
5Step 5: Solve the Quadratic Equation
The complete factorization using the quadratic is:\[ (x + 1)(x^2 - 4) \]Substitute back and solve:\(x^2 - 4 = 0\) gives roots \(x = 2\) and \(x = -2\).
6Step 6: List All Intercepts
The roots found are \(x = -1\), \(x = 2\), and \(x = -2\). Hence, the x-intercepts of the polynomial \(f(x)\) are at \((-1, 0)\), \((2, 0)\), and \((-2, 0)\).
Key Concepts
Rational Root TheoremPolynomial DivisionFactoring PolynomialsX-Intercepts
Rational Root Theorem
The Rational Root Theorem is a useful tool when working with polynomial functions, especially when trying to identify possible rational roots (or x-intercepts). This theorem states that, for a polynomial equation of the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\), any potential rational root \(\frac{p}{q}\) must have a numerator \(p\) that is a factor of the constant term \(a_0\) and a denominator \(q\) that is a factor of the leading coefficient \(a_n\).
In the case of the polynomial \(f(x)=x^{3}+x^{2}-4x-4\), the constant term is \(-4\) and the leading coefficient is \(1\). Thus, the possible rational roots to test are \(\pm 1, \pm 2, \pm 4\). After testing these values, only \(x = -1\) is found to be a root, making it integral in further factoring the polynomial.
In the case of the polynomial \(f(x)=x^{3}+x^{2}-4x-4\), the constant term is \(-4\) and the leading coefficient is \(1\). Thus, the possible rational roots to test are \(\pm 1, \pm 2, \pm 4\). After testing these values, only \(x = -1\) is found to be a root, making it integral in further factoring the polynomial.
Polynomial Division
Polynomial division is akin to the long division method used in arithmetic. It is necessary for breaking down a complex polynomial into simpler parts, typically by dividing by a factor of the polynomial that has been verified as a root. This process can either be carried out by long division or by synthetic division, a shortcut for certain types of polynomials involving one root.
For the polynomial \(x^{3}+x^{2}-4x-4\) and known root \(x = -1\), we can directly transform the full expression to \((x + 1)\), due to \(x - (-1)\), and use this in polynomial division. After performing polynomial division on \(x^{3}+x^{2}-4x-4\) by \(x + 1\), the quotient is \(x^2 - 4\), helping us to factor the overall expression further.
For the polynomial \(x^{3}+x^{2}-4x-4\) and known root \(x = -1\), we can directly transform the full expression to \((x + 1)\), due to \(x - (-1)\), and use this in polynomial division. After performing polynomial division on \(x^{3}+x^{2}-4x-4\) by \(x + 1\), the quotient is \(x^2 - 4\), helping us to factor the overall expression further.
Factoring Polynomials
Factoring polynomials involves rewriting the polynomial as a product of its simpler factors. This step is crucial in finding the roots of the equation. After using polynomial division or identifying a known root, the next step is to factor any remaining quadratic or lower-order polynomials.
From our division of \(x^{3}+x^{2}-4x-4\) by \(x + 1\), we obtained \(x^2 - 4\). Recognizing this as a difference of squares, it can be factored further to \((x - 2)(x + 2)\). Thus, the entire polynomial factors to \((x + 1)(x - 2)(x + 2)\). This expression makes it straightforward to find all the x-intercepts or roots of the original polynomial.
From our division of \(x^{3}+x^{2}-4x-4\) by \(x + 1\), we obtained \(x^2 - 4\). Recognizing this as a difference of squares, it can be factored further to \((x - 2)(x + 2)\). Thus, the entire polynomial factors to \((x + 1)(x - 2)(x + 2)\). This expression makes it straightforward to find all the x-intercepts or roots of the original polynomial.
X-Intercepts
Finding x-intercepts is one of the primary goals when working with polynomial functions. The intercepts represent the points where the polynomial crosses the x-axis, indicating the roots of the equation. Each intercept corresponds to a root of the polynomial.
After factoring the polynomial \(f(x)=x^{3}+x^{2}-4x-4\) into \((x + 1)(x - 2)(x + 2)\), we can easily see the x-intercepts:
After factoring the polynomial \(f(x)=x^{3}+x^{2}-4x-4\) into \((x + 1)(x - 2)(x + 2)\), we can easily see the x-intercepts:
- \(x = -1\) from \(x + 1 = 0\)
- \(x = 2\) from \(x - 2 = 0\)
- \(x = -2\) from \(x + 2 = 0\)
Other exercises in this chapter
Problem 15
For the following exercises, identify the function as a power function, a polynomial function, or neither. $$ x\left(4-x^{2}\right)(2 x+1) $$
View solution Problem 15
For the following exercises, use synthetic division to find the quotient. $$ \left(2 x^{3}-6 x^{2}-7 x+6\right) \div(x-4) $$
View solution Problem 15
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=
View solution Problem 15
Find the degree and leading coefficient for the given polynomial. $$x\left(4-x^{2}\right)(2 x+1)$$
View solution