Problem 56
Question
State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The constant term of \(p(x) q(x)\).
Step-by-Step Solution
Verified Answer
Answer: The constant term of the product of the two given polynomials p(x) and q(x) is -6.
1Step 1: List given information
We are given the following information:
- \(p(x)\) is a polynomial of degree 5 with constant term 3
- \(q(x)\) is a polynomial of degree 8 with constant term -2
2Step 2: Multiply the constant terms
To find the constant term of the product p(x)q(x), multiply the constant terms of both polynomials:
Constant term of p(x)q(x) = (Constant term of p(x)) × (Constant term of q(x))
3Step 3: Calculate the constant term of p(x)q(x)
According to the given information, the constant terms of p(x) and q(x) are 3 and -2, respectively. Now, multiply these constant terms:
Constant term of p(x)q(x) = 3 × (-2) = -6
4Step 4: State the answer
The constant term of the product of the two polynomials p(x) and q(x) is -6.
Key Concepts
Constant TermPolynomial MultiplicationDegree of a Polynomial
Constant Term
In polynomials, the constant term is the term that does not contain any variable. It stands alone as a number. For example, in the polynomial \(2x^3 + 4x^2 + 5\), the constant term is 5.
This is important because when we multiply polynomials together, even though many terms are created and eliminated, the constant term remains a crucial part we can easily determine.
To find the constant term in a product of polynomials, like in our exercise, you multiply the constant terms of each polynomial together. For instance, if we have \(p(x)\) with a constant term of 3, and \(q(x)\) with a constant term of -2, then the constant term of their product, \(p(x)q(x)\), is calculated by multiplying 3 by -2, which results in -6.
This is important because when we multiply polynomials together, even though many terms are created and eliminated, the constant term remains a crucial part we can easily determine.
To find the constant term in a product of polynomials, like in our exercise, you multiply the constant terms of each polynomial together. For instance, if we have \(p(x)\) with a constant term of 3, and \(q(x)\) with a constant term of -2, then the constant term of their product, \(p(x)q(x)\), is calculated by multiplying 3 by -2, which results in -6.
- The constant terms can be identified directly from the polynomial without any calculations.
- They help in deriving specific characteristics of the polynomial form.
Polynomial Multiplication
Polynomial multiplication involves multiplying two or more polynomials together to form a new polynomial. The process follows the distributive property, where each term in the first polynomial is multiplied by each term in the second polynomial, and then you combine like terms.
During multiplication, it's essential to pay attention to the degrees of the resulting terms and the sign of each term. For instance, multiplying a term with a positive coefficient by another with a negative coefficient will yield a negative result. A methodical approach is needed to organize computations, especially for large polynomials.
When multiplying polynomials \(p(x)\) and \(q(x)\), we focus initially on the constant terms and degrees. This helps us identify important properties without dealing with the complexity of all terms.
During multiplication, it's essential to pay attention to the degrees of the resulting terms and the sign of each term. For instance, multiplying a term with a positive coefficient by another with a negative coefficient will yield a negative result. A methodical approach is needed to organize computations, especially for large polynomials.
When multiplying polynomials \(p(x)\) and \(q(x)\), we focus initially on the constant terms and degrees. This helps us identify important properties without dealing with the complexity of all terms.
- Always apply the distributive property while ensuring the combination of like terms.
- Note the signs which will influence addition and subtraction of the terms.
Degree of a Polynomial
The degree of a polynomial is one of its most essential characteristics. It is defined as the highest power of the variable in the polynomial expression. For example, in \(3x^5 + 4x^3 + 2\), the degree is 5, because the highest power of x is 5.
Knowing the degree of a polynomial is vital when analyzing or performing operations like addition or multiplication. For multiplication, the degree of the resulting polynomial is typically the sum of the degrees of the multiplied polynomials.
In the problem given, \(p(x)\) is a polynomial with degree 5, and \(q(x)\) is of degree 8. Hence, the degree of the product \(p(x)q(x)\) will be 5 + 8 = 13. This helps predict the behavior and complexity of the polynomial product.
Knowing the degree of a polynomial is vital when analyzing or performing operations like addition or multiplication. For multiplication, the degree of the resulting polynomial is typically the sum of the degrees of the multiplied polynomials.
In the problem given, \(p(x)\) is a polynomial with degree 5, and \(q(x)\) is of degree 8. Hence, the degree of the product \(p(x)q(x)\) will be 5 + 8 = 13. This helps predict the behavior and complexity of the polynomial product.
- The highest degree guides the graph shape and end behavior of the polynomial function.
- It determines the number of roots that the polynomial equation can have.
Other exercises in this chapter
Problem 54
Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a
View solution Problem 55
If the following product of two polynomials, $$\left(3 t^{2}-7 t-2\right)\left(4 t^{3}-3 t^{2}+5\right)$$ is written in standard form, what are the constant and
View solution Problem 57
State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The de
View solution Problem 58
State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The de
View solution