When $\left(4x^2 - 2x - 3\right$\] And $(x - 3$\] Are Multiplied, The Product Is $4x^3 - 14x^2 + Cx + 9$.The Value Of $c$ Is $\square$.

Introduction

In algebra, polynomial multiplication is a fundamental concept that involves multiplying two or more polynomials to obtain a new polynomial. In this article, we will explore a specific problem where two polynomials are multiplied, and we need to find the value of a coefficient in the resulting polynomial.

Problem Statement

Given two polynomials:

$$\left(4x^2 - 2x - 3\right)$$

and

$$(x - 3)$$

when multiplied, the product is:

$$4x^3 - 14x^2 + cx + 9$$

Our goal is to find the value of the coefficient $c$ in the resulting polynomial.

Step 1: Multiply the Two Polynomials

To find the product of the two polynomials, we need to multiply each term of the first polynomial by each term of the second polynomial and then combine like terms.

Let's start by multiplying the first polynomial by the first term of the second polynomial, which is $x$:

$$\left(4x^2 - 2x - 3\right) \cdot x = 4x^3 - 2x^2 - 3x$$

Next, we multiply the first polynomial by the second term of the second polynomial, which is $-3$:

$$\left(4x^2 - 2x - 3\right) \cdot (-3) = -12x^2 + 6x + 9$$

Now, we add the two products together to get the final product:

$$4x^3 - 2x^2 - 3x - 12x^2 + 6x + 9 = 4x^3 - 14x^2 + 3x + 9$$

Step 2: Compare the Resulting Polynomial with the Given Product

We are given that the product of the two polynomials is:

$$4x^3 - 14x^2 + cx + 9$$

Comparing this with the polynomial we obtained in Step 1, we can see that the coefficients of the $x^3$ and $x^2$ terms are the same. However, the coefficient of the $x$ term in our polynomial is $3$, while the coefficient of the $x$ term in the given product is $c$.

Step 3: Solve for the Value of c

Since the coefficients of the $x$ term in our polynomial and the given product are different, we can set up an equation to solve for the value of $c$:

$$3 = c$$

Therefore, the value of $c$ is $\boxed{3}$.

Conclusion

In this article, we solved a polynomial multiplication problem to find the value of a coefficient in the resulting polynomial. We multiplied two polynomials, compared the resulting polynomial with the given product, and solved for the value of $c$. The value of $c$ is $\boxed{3}$.

Additional Examples

To practice solving polynomial multiplication problems, try the following examples:

• Multiply the polynomials $(x^2 + 2x - 1)$ and $(x + 1)$ to find the value of the coefficient in the resulting polynomial.
• Multiply the polynomials $(2x^2 - 3x + 1)$ and $(x - 2)$ to find the value of the coefficient in the resulting polynomial.

References

• Polynomial Multiplication
• Algebraic Manipulation

Glossary

• Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
• Coefficient: A number that is multiplied by a variable in a polynomial.
• Term: A single part of a polynomial, consisting of a variable and a coefficient.
• Like Terms: Terms that have the same variable and exponent.
  Frequently Asked Questions (FAQs) about Polynomial Multiplication ====================================================================

Q: What is polynomial multiplication?

A: Polynomial multiplication is the process of multiplying two or more polynomials to obtain a new polynomial. It involves multiplying each term of one polynomial by each term of the other polynomial and then combining like terms.

Q: How do I multiply two polynomials?

A: To multiply two polynomials, you need to multiply each term of one polynomial by each term of the other polynomial and then combine like terms. You can use the distributive property to multiply each term of one polynomial by each term of the other polynomial.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms. In the context of polynomial multiplication, it means that you can multiply each term of one polynomial by each term of the other polynomial.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable and exponent. For example, if you have two terms with the same variable and exponent, you can add their coefficients to get the final term.

Q: What is the difference between polynomial multiplication and polynomial addition?

A: Polynomial multiplication involves multiplying two or more polynomials to obtain a new polynomial, while polynomial addition involves adding two or more polynomials to obtain a new polynomial.

Q: Can I multiply a polynomial by a constant?

A: Yes, you can multiply a polynomial by a constant. When you multiply a polynomial by a constant, you multiply each term of the polynomial by the constant.

Q: Can I multiply a polynomial by a variable?

A: Yes, you can multiply a polynomial by a variable. When you multiply a polynomial by a variable, you multiply each term of the polynomial by the variable.

Q: How do I multiply a polynomial by a binomial?

A: To multiply a polynomial by a binomial, you need to multiply each term of the polynomial by each term of the binomial and then combine like terms.

Q: How do I multiply a polynomial by a trinomial?

A: To multiply a polynomial by a trinomial, you need to multiply each term of the polynomial by each term of the trinomial and then combine like terms.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It involves multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: Can I use the FOIL method to multiply a polynomial by a binomial?

A: Yes, you can use the FOIL method to multiply a polynomial by a binomial. However, you need to be careful when multiplying the polynomial by the binomial, as the FOIL method only works for binomials.

Q: What is the difference between the FOIL method and the distributive property?

A: The FOIL method is a technique used to multiply two binomials, while the distributive property is a mathematical property that states that a single term can be distributed to multiple terms.

Q: Can I use the distributive property to multiply a polynomial by a binomial?

A: Yes, you can use the distributive property to multiply a polynomial by a binomial. However, the FOIL method is a more efficient way to multiply two binomials.

Q: How do I check my work when multiplying polynomials?

A: To check your work when multiplying polynomials, you need to multiply the polynomials again and compare the result with the original product. If the two products are the same, then your work is correct.

Q: What are some common mistakes to avoid when multiplying polynomials?

A: Some common mistakes to avoid when multiplying polynomials include:

• Forgetting to multiply each term of one polynomial by each term of the other polynomial
• Forgetting to combine like terms
• Making errors when multiplying the polynomials
• Not checking the work

Q: How do I multiply polynomials with negative exponents?

A: To multiply polynomials with negative exponents, you need to follow the same steps as multiplying polynomials with positive exponents. However, you need to be careful when multiplying the terms with negative exponents.

Q: Can I multiply polynomials with fractional exponents?

A: Yes, you can multiply polynomials with fractional exponents. However, you need to be careful when multiplying the terms with fractional exponents.

Q: How do I multiply polynomials with complex numbers?

A: To multiply polynomials with complex numbers, you need to follow the same steps as multiplying polynomials with real numbers. However, you need to be careful when multiplying the terms with complex numbers.

Q: Can I use a calculator to multiply polynomials?

A: Yes, you can use a calculator to multiply polynomials. However, you need to be careful when using a calculator, as it may make mistakes.

Q: How do I multiply polynomials with multiple variables?

A: To multiply polynomials with multiple variables, you need to follow the same steps as multiplying polynomials with one variable. However, you need to be careful when multiplying the terms with multiple variables.

Q: Can I use the FOIL method to multiply polynomials with multiple variables?

A: No, you cannot use the FOIL method to multiply polynomials with multiple variables. The FOIL method only works for binomials.

Q: How do I multiply polynomials with coefficients?

A: To multiply polynomials with coefficients, you need to follow the same steps as multiplying polynomials without coefficients. However, you need to be careful when multiplying the coefficients.

Q: Can I use the distributive property to multiply polynomials with coefficients?

A: Yes, you can use the distributive property to multiply polynomials with coefficients. However, the FOIL method is a more efficient way to multiply two binomials.

Q: How do I multiply polynomials with variables and coefficients?

A: To multiply polynomials with variables and coefficients, you need to follow the same steps as multiplying polynomials with variables. However, you need to be careful when multiplying the coefficients.

Q: Can I use the FOIL method to multiply polynomials with variables and coefficients?

A: No, you cannot use the FOIL method to multiply polynomials with variables and coefficients. The FOIL method only works for binomials.

Q: How do I multiply polynomials with multiple terms?

A: To multiply polynomials with multiple terms, you need to follow the same steps as multiplying polynomials with one term. However, you need to be careful when multiplying the terms.

Q: Can I use the distributive property to multiply polynomials with multiple terms?

A: Yes, you can use the distributive property to multiply polynomials with multiple terms. However, the FOIL method is a more efficient way to multiply two binomials.

Q: How do I multiply polynomials with like terms?

A: To multiply polynomials with like terms, you need to follow the same steps as multiplying polynomials without like terms. However, you need to be careful when multiplying the like terms.

Q: Can I use the FOIL method to multiply polynomials with like terms?

A: No, you cannot use the FOIL method to multiply polynomials with like terms. The FOIL method only works for binomials.

Q: How do I multiply polynomials with unlike terms?

A: To multiply polynomials with unlike terms, you need to follow the same steps as multiplying polynomials without unlike terms. However, you need to be careful when multiplying the unlike terms.

Q: Can I use the distributive property to multiply polynomials with unlike terms?

A: Yes, you can use the distributive property to multiply polynomials with unlike terms. However, the FOIL method is a more efficient way to multiply two binomials.

Q: How do I multiply polynomials with negative terms?

A: To multiply polynomials with negative terms, you need to follow the same steps as multiplying polynomials without negative terms. However, you need to be careful when multiplying the negative terms.

Q: Can I use the FOIL method to multiply polynomials with negative terms?

A: No, you cannot use the FOIL method to multiply polynomials with negative terms. The FOIL method only works for binomials.

Q: How do I multiply polynomials with fractional terms?

A: To multiply polynomials with fractional terms, you need to follow the same steps as multiplying polynomials without fractional terms. However, you need to be careful when multiplying the fractional terms.

Q: Can I use the FOIL method to multiply polynomials with fractional terms?

A: No, you cannot use the FOIL method to multiply polynomials with fractional terms. The FOIL method only works for binomials.

Q: How do I multiply polynomials with complex terms?

A: To multiply polynomials with complex terms, you need to follow the same steps as multiplying polynomials without complex terms. However, you need to be careful when multiplying the complex terms.

Q: Can I use the FOIL method to multiply polynomials with complex terms?

A: No, you cannot use the FOIL method to multiply polynomials with