Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 59
A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$
Short Answer
Expert verified
Zeros are \(x = -\frac{1}{2}\) and \(x = 2\) (double root). Graph shows end behavior from \(-\infty\) to \(\infty\).
Step by step solution
01
Analyze the Polynomial Degree
The given polynomial function, \(P(x) = 2x^3 - 7x^2 + 4x + 4\), is a cubic polynomial because the highest power of \(x\) is 3. A cubic polynomial can have up to 3 real roots.
02
Determine the Possible Rational Zeros
By the Rational Root Theorem, any rational zero, \( \frac{p}{q} \), of \( P(x) \) must be a factor of the constant term (4) divided by a factor of the leading coefficient (2). The possible rational zeros are \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2}\).
03
Test Each Possible Rational Zero
Evaluate \(P(x)\) at each possible rational zero. Check \(P(1)\), \(P(-1)\), \(P(2)\), etc., to see if any of these values make the polynomial equal to zero. Upon testing:- \(P(1) = 2(1)^3 - 7(1)^2 + 4(1) + 4 = 3\) (not zero)- \(P(-1) = 2(-1)^3 - 7(-1)^2 + 4(-1) + 4 = -9\) (not zero)- \(P(2) = 2(2)^3 - 7(2)^2 + 4(2) + 4 = 0\) (zero, so \(x=2\) is a root)Now divide the original polynomial by \(x-2\).
04
Perform Synthetic Division
Perform synthetic division of \(P(x)\) by \(x - 2\):Dividing yields \(2x^2 - 3x - 2\). The quotient is a quadratic polynomial.
05
Factor the Quotient Polynomial
Factor the quadratic polynomial \(2x^2 - 3x - 2\). This can be written as \((2x + 1)(x - 2)\), giving the complete factorization of \(P(x)\) as \(P(x) = (x - 2)(2x^2 - 3x - 2)\).
06
Solve for Remaining Zeros
Set each factor to zero to find the remaining zeros:- For \(2x + 1 = 0\), solve to get \(x = -\frac{1}{2}\).- \(x = 2\) was found earlier but appears as a repeated root.
07
Sketch the Graph Using Zeros and End Behavior
Sketch the graph of \(P(x)\):- Zeros at \(x = -\frac{1}{2}, 2\) (2 as a double root).- End behavior: As \(x \to \infty\), \(P(x) \to \infty\), and as \(x \to -\infty\), \(P(x) \to -\infty\).- The graph crosses the x-axis at \(x = -\frac{1}{2}\) and touches the x-axis at \(x=2\) without crossing due to the double root.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Root Theorem
The Rational Root Theorem is a powerful tool that helps us predict the possible rational zeros of a polynomial.
When faced with a polynomial, it can be daunting to guess what inputs (roots) will make the polynomial equal to zero.
The Rational Root Theorem simplifies this by offering a list of potential candidates. Here's how it works:
When faced with a polynomial, it can be daunting to guess what inputs (roots) will make the polynomial equal to zero.
The Rational Root Theorem simplifies this by offering a list of potential candidates. Here's how it works:
- First, identify the constant term and the leading coefficient of the polynomial. In the polynomial \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), the constant term is \( 4 \) and the leading coefficient is \( 2 \).
- According to the theorem, any rational zero \( \frac{p}{q} \) must have \( p \), a factor of the constant term \( 4 \), and \( q \), a factor of the leading coefficient \( 2 \).
- The factors of \( 4 \) are \( \pm 1, \pm 2, \pm 4 \) and the factors of \( 2 \) are \( \pm 1, \pm 2 \).
- This provides a list of possible rational roots: \( \pm 1, \pm 2, \pm 4, \pm \frac{1}{2} \).
cubic polynomial
A cubic polynomial is a type of polynomial of degree three.
It takes the general form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \).
In our example, \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), clearly fits this form.Key characteristics of cubic polynomials include:
It takes the general form \( ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \).
In our example, \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), clearly fits this form.Key characteristics of cubic polynomials include:
- They can have up to three real roots (solutions) because it's a third-order polynomial.
- The graph of a cubic polynomial typically has a characteristic "S" shape, representing its three degrees and inflection point.
- As \( x \to \infty \), a cubic function \( ax^3 \) tends towards \( \, ext{either}\, \pm \infty \) depending on the leading coefficient \( a \).
synthetic division
Synthetic division is a short method for dividing a polynomial by a linear binomial of the form \( x - r \).
It's especially useful when applying the Rational Root Theorem to test potential roots.
Here's a brief overview of how synthetic division works:
This efficient process makes it easier to factor polynomials and simplify root finding.
It's especially useful when applying the Rational Root Theorem to test potential roots.
Here's a brief overview of how synthetic division works:
- First, write down the coefficients of the polynomial. For \( P(x) = 2x^3 - 7x^2 + 4x + 4 \), write the coefficients: \( 2, -7, 4, 4 \).
- Then, write the value of \( r \) (the test root) off to the left. Let's take \( r = 2 \) as found in our example.
- Perform synthetic division by bringing down the leading coefficient and repeatedly multiplying and adding.
- This division process produces a new polynomial (the quotient) and determines if \( x - r \) is a factor by checking the remainder.
This efficient process makes it easier to factor polynomials and simplify root finding.
