Problem 37
Question
Determine together \(q(x)\) is a factor of \(p(x)\) Here, \(p(x)\) is the first polynomial and \(q(x)\) is the second polynomial. justify your answer. $$3 x^{3}-48 x-4 x^{2}+64 ; x+4$$
Step-by-Step Solution
Verified Answer
After substituting \(x = -4\) in \(p(x)\), the result comes out to be zero. Therefore, \(x+4\) is a factor of the polynomial \(p(x)\).
1Step 1: Apply Remainder Theorem
The Remainder Theorem states that if a polynomial \(p(x)\) is divided by \(x-c\), the remainder is \(p(c)\). So, if \(q(x) = x+4\) is a factor, then substituting \(x = -4\) in \(p(x)\) should yield zero.
2Step 2: Substitute x in \(p(x)\)
Substitute \(x = -4\) into \(p(x) = 3x^{3}-48x-4x^{2}+64\).
3Step 3: Evaluate the Expression
Calculate the value of \(p(-4)\). If it is zero, then \(x+4\) is the factor of \(p(x)\), otherwise, it's not.
Key Concepts
Remainder TheoremFactors of PolynomialsEvaluating PolynomialsPolynomial Expressions
Remainder Theorem
The Remainder Theorem is an important concept when working with polynomials. This theorem provides a quick way to find the remainder of dividing a polynomial, \( p(x) \), by a linear polynomial of the form \( x - c \). According to the theorem, the remainder you get when \( p(x) \) is divided by \( x-c \) is equal to \( p(c) \).
This means, if \( p(c) = 0 \), the polynomial \( x-c \) is a factor of \( p(x) \). The theorem is particularly useful because it allows you to determine if \( x-c \) is a factor without performing long division on the polynomials.
In practice, this means substituting \( c \) into the polynomial and simplifying to see if the result is zero. For example, for \( p(x) = 3x^{3}-48x-4x^{2}+64 \) and \( q(x) = x+4 \), replace \( x \) with \(-4\) (since \( x+4 \) can be rewritten in the form \( x - (-4) \)) and calculate \( p(-4) \). If \( p(-4) = 0 \), then \( x+4 \) is a factor of \( p(x) \).
This means, if \( p(c) = 0 \), the polynomial \( x-c \) is a factor of \( p(x) \). The theorem is particularly useful because it allows you to determine if \( x-c \) is a factor without performing long division on the polynomials.
In practice, this means substituting \( c \) into the polynomial and simplifying to see if the result is zero. For example, for \( p(x) = 3x^{3}-48x-4x^{2}+64 \) and \( q(x) = x+4 \), replace \( x \) with \(-4\) (since \( x+4 \) can be rewritten in the form \( x - (-4) \)) and calculate \( p(-4) \). If \( p(-4) = 0 \), then \( x+4 \) is a factor of \( p(x) \).
Factors of Polynomials
Factors of a polynomial are the expressions that, when multiplied together, give the original polynomial. Knowing the factors of a polynomial is crucial for solving polynomial equations carefully and efficiently.
When looking to identify the factors of a polynomial, such as \( 3x^{3}-48x-4x^{2}+64 \), we can use methods like the Remainder Theorem, Factor Theorem, or polynomial long division. Each factor corresponds to a root of the polynomial when the polynomial is set to zero.
When looking to identify the factors of a polynomial, such as \( 3x^{3}-48x-4x^{2}+64 \), we can use methods like the Remainder Theorem, Factor Theorem, or polynomial long division. Each factor corresponds to a root of the polynomial when the polynomial is set to zero.
- If a polynomial \( p(x) \) has a factor \( x - a \), then \( a \) is a root of the polynomial.
- To confirm if a polynomial like \( x+4 \) is a factor, we can verify if substituting \( x = -4 \) into the polynomial results in zero, using the Remainder Theorem.
Evaluating Polynomials
Evaluating a polynomial involves calculating its value for a specific value of the variable \( x \). This process helps determine whether a given number is a root of the polynomial.
Suppose we have \( p(x) = 3x^{3} - 48x - 4x^{2} + 64 \) and we want to find \( p(-4) \). To evaluate, substitute \( x = -4 \) into the polynomial and perform the calculations:
Suppose we have \( p(x) = 3x^{3} - 48x - 4x^{2} + 64 \) and we want to find \( p(-4) \). To evaluate, substitute \( x = -4 \) into the polynomial and perform the calculations:
- Calculate each term separately: \( 3(-4)^{3} \), \(-4(-4)^{2} \), \(-48(-4) \), and \(64 \).
- Sum up all the results from each calculation.
Polynomial Expressions
Polynomial expressions are algebraic expressions composed of variables raised to whole-number exponents and their coefficients. They form the basis for polynomial functions, equations, and various mathematical problems.
Polynomial expressions range from simple monomials like \( 2x \) to more complicated polynomials like \( 3x^{3} - 48x - 4x^{2} + 64 \). Each term in a polynomial has a coefficient (a constant multiplied by the variable term) and a degree (the highest power of the variable).
Polynomial expressions range from simple monomials like \( 2x \) to more complicated polynomials like \( 3x^{3} - 48x - 4x^{2} + 64 \). Each term in a polynomial has a coefficient (a constant multiplied by the variable term) and a degree (the highest power of the variable).
- In \( 3x^{3} \), the coefficient is 3 and the degree of the term is 3.
- The structure of polynomial expressions allows flexibility in mathematical computation, such as manipulation through addition, subtraction, division, and multiplication.
Other exercises in this chapter
Problem 37
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For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multipliciti
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