Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 16
If \((x+1)\) and \((x-1)\) are the factors of \(a x^{3}+b x^{2}+c x+d\), then which of the following is true? (1) \(\mathrm{a}+\mathrm{b}=0\) (2) \(\mathrm{b}+\mathrm{c}=0\) (3) \(\mathrm{b}+\mathrm{d}=0\) (4) None of these
Short Answer
Expert verified
(1) a + b = 0
(2) b + c = 0
(3) b + d = 0
Answer: (3) b + d = 0
Step by step solution
01
Expand the Polynomial
Since \((x+1)\) and \((x-1)\) are the factors of the given polynomial, we can write the polynomial as \((x+1)(x-1)(kx+m)\), where k and m are constants which will allow the polynomial to match the given coefficients. Firstly, we need to expand the given factorized polynomial:
\((x+1)(x-1)(kx+m) = (x^2-1)(kx+m)\)
02
Distribute the Terms
Now distribute each term from \((kx+m)\) to \((x^2-1)\):
\((kx+m)(x^2-1) = kx(x^2-1) + m(x^2-1)\)
03
Simplify the Expression
Expand the terms:
\(kx(x^2-1) + m(x^2-1) = kx^3 - kx + mx^2 - m\)
Now, we can write the entire polynomial as:
\(P(x) = ax^3 + bx^2 + cx + d = kx^3 + mx^2 - kx - m\)
04
Compare Coefficients
Compare the coefficients in \(P(x)\) with the given polynomial:
\(a = k\)
\(b = m\)
\(c = -k\)
\(d = -m\)
Since we are looking for a relationship between the coefficients a, b, c, and d, let's analyze the given statements:
05
Statement 1
If statement 1 is true, then \(a+b=0\). Substitute the coefficients:
\(k + m = 0\)
06
Statement 2
If statement 2 is true, then \(b+c=0\). Substitute the coefficients:
\(m - k = 0\)
07
Statement 3
If statement 3 is true, then \(b+d=0\). Substitute the coefficients:
\(m - m = 0\)
We can see that:
1. Statement 1 is true for k=-m, which is not necessarily true for any cubic polynomial.
2. Statement 2 is true for k=m, which may turn out to be true after further analysis.
3. Statement 3 is always true.
08
Conclusion
From our analysis, we can see that statement 3 is true for any given cubic polynomial with the factors \((x+1)\) and \((x-1)\). Thus, the correct answer is:
(3) \(b+d=0\)
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.
Cubic Polynomial
A cubic polynomial is an algebraic expression of the form \( ax^3 + bx^2 + cx + d \), where \( a eq 0 \). This type of polynomial is called 'cubic' because the highest exponent is 3. Cubic polynomials play a fundamental role in various mathematical computations since they can describe complex relationships and models.
Understanding the structure of a cubic polynomial is important for factorization, solving equations, and modeling different scenarios. Each term of a cubic polynomial contributes to its shape when graphed, and the coefficients \( a, b, c, \) and \( d \) each have their own influence. For instance:
Understanding the structure of a cubic polynomial is important for factorization, solving equations, and modeling different scenarios. Each term of a cubic polynomial contributes to its shape when graphed, and the coefficients \( a, b, c, \) and \( d \) each have their own influence. For instance:
- \( a \) affects the width and direction of the curve.
- \( b \) modifies the shape and location of the inflection point.
- \( c \) adjusts the asymmetry of the curve.
- \( d \) stands for the y-intercept, shifting the graph vertically.
Coefficients Comparison
Coefficients comparison is a method used to find the values of unknown parameters in a polynomial equation. By comparing corresponding coefficients from both sides of a polynomial equation, one can derive equations that involve these unknown parameters. In this particular exercise, you learned how to use coefficients comparison to relate the given polynomial \( ax^3 + bx^2 + cx + d \) to its factorized counterpart.
In this process, notice how we substitute the coefficients from our expanded polynomial \( kx^3 + mx^2 - kx - m \) into \( ax^3 + bx^2 + cx + d \):
In this process, notice how we substitute the coefficients from our expanded polynomial \( kx^3 + mx^2 - kx - m \) into \( ax^3 + bx^2 + cx + d \):
- \( a = k \)
- \( b = m \)
- \( c = -k \)
- \( d = -m \)
Factor Theorem
The Factor Theorem is a principal that provides a straightforward way to ascertain whether a linear expression is a factor of a given polynomial. According to the theorem, for a polynomial \( f(x) \), if \( f(a) = 0 \), then \( (x-a) \) is a factor of \( f(x) \). In this exercise, you were given two factors, \((x+1) \) and \( (x-1) \), meaning "\( x = -1 \)" and "\( x = 1 \)" are roots of the cubic polynomial \( ax^3 + bx^2 + cx + d \).
Using the Factor Theorem in this context, we factor the cubic polynomial into \((x+1)(x-1)(kx+m)\) to demonstrate that \((x+1)\) and \((x-1)\) are factors.
The Factor Theorem is incredibly helpful for:
Using the Factor Theorem in this context, we factor the cubic polynomial into \((x+1)(x-1)(kx+m)\) to demonstrate that \((x+1)\) and \((x-1)\) are factors.
The Factor Theorem is incredibly helpful for:
- **Finding roots of polynomials:** With roots like \(-1\) and \(1\), it becomes easier to factor and solve polynomials efficiently.
- **Simplifying polynomial expressions:** Breaking down a polynomial into simpler factors allows easier manipulation and solution finding.
- **Helpful in higher-degree polynomial factorization:** Understanding which values cause the polynomial to become zero makes it easier to attempt further factorization into simpler components.
